| L(s) = 1 | − 2·2-s + 10·3-s + 6·4-s + 10·5-s − 20·6-s + 4·8-s + 79·9-s − 20·10-s − 66·11-s + 60·12-s − 20·13-s + 100·15-s + 24·16-s + 70·17-s − 158·18-s + 140·19-s + 60·20-s + 132·22-s + 16·23-s + 40·24-s + 25·25-s + 40·26-s + 830·27-s − 516·29-s − 200·30-s − 20·31-s + 120·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.92·3-s + 3/4·4-s + 0.894·5-s − 1.36·6-s + 0.176·8-s + 2.92·9-s − 0.632·10-s − 1.80·11-s + 1.44·12-s − 0.426·13-s + 1.72·15-s + 3/8·16-s + 0.998·17-s − 2.06·18-s + 1.69·19-s + 0.670·20-s + 1.27·22-s + 0.145·23-s + 0.340·24-s + 1/5·25-s + 0.301·26-s + 5.91·27-s − 3.30·29-s − 1.21·30-s − 0.115·31-s + 0.662·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.282136060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.282136060\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T - p T^{2} - 5 p^{2} T^{3} - 15 p^{2} T^{4} - 5 p^{5} T^{5} - p^{7} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 - 5 T - 2 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 p T + 71 p T^{2} + 498 p^{2} T^{3} + 43068 p^{2} T^{4} + 498 p^{5} T^{5} + 71 p^{7} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 10 T + 19 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 70 T - 103 p T^{2} + 222250 T^{3} - 3975468 T^{4} + 222250 p^{3} T^{5} - 103 p^{7} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 140 T + 5382 T^{2} - 70000 T^{3} + 20669243 T^{4} - 70000 p^{3} T^{5} + 5382 p^{6} T^{6} - 140 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T - 15518 T^{2} + 136960 T^{3} + 97668435 T^{4} + 136960 p^{3} T^{5} - 15518 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 258 T + 40075 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 20 T - 19682 T^{2} - 790000 T^{3} - 496133357 T^{4} - 790000 p^{3} T^{5} - 19682 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 328 T + 678 T^{2} + 1836800 T^{3} + 3413714075 T^{4} + 1836800 p^{3} T^{5} + 678 p^{6} T^{6} + 328 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 300 T + 50342 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 30 T - 189371 T^{2} - 521250 T^{3} + 25330397412 T^{4} - 521250 p^{3} T^{5} - 189371 p^{6} T^{6} + 30 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 540 T + 79346 T^{2} - 46170000 T^{3} - 20525133813 T^{4} - 46170000 p^{3} T^{5} + 79346 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 380 T - 284858 T^{2} - 7030000 T^{3} + 112425169323 T^{4} - 7030000 p^{3} T^{5} - 284858 p^{6} T^{6} - 380 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 1080 T + 460438 T^{2} - 272160000 T^{3} + 182111332683 T^{4} - 272160000 p^{3} T^{5} + 460438 p^{6} T^{6} - 1080 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 468 T - 335882 T^{2} - 21818160 T^{3} + 151587971355 T^{4} - 21818160 p^{3} T^{5} - 335882 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1056 T + 949550 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 860 T + 216666 T^{2} - 219386000 T^{3} - 165591231133 T^{4} - 219386000 p^{3} T^{5} + 216666 p^{6} T^{6} + 860 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 2 p T + 364789 T^{2} - 2651806 p T^{3} - 139914650492 T^{4} - 2651806 p^{4} T^{5} + 364789 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 40 T + 703974 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 240 T + 221662 T^{2} - 377760000 T^{3} - 510671165517 T^{4} - 377760000 p^{3} T^{5} + 221662 p^{6} T^{6} + 240 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 1630 T + 2133171 T^{2} + 1630 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.315199211438311139449602733813, −8.141125406824900167206485656212, −7.74625580802221601183162688047, −7.66705103341604041720063058618, −7.43698690249721245608208939815, −7.20358123533573422575507075440, −6.88389736978924424688416294339, −6.78332281927936503136572651972, −6.39369944272688583628923700509, −5.71209538941045048665919269835, −5.49563093099753924726975872492, −5.43998430846459827254227490325, −5.09891356320569881797998139412, −4.63193146403438664272575340892, −4.53518390951766799385481065313, −3.72519515941886934074736349436, −3.57268462936743338421946780884, −3.27113235036049976245479313914, −2.87427635825839520143647599459, −2.65213095962528783073089481806, −2.33229638549031083540767781806, −1.59463382226284353460502792814, −1.46433900098329622080759239298, −1.41752187347762205997632838444, −0.33844029377744374373785922247,
0.33844029377744374373785922247, 1.41752187347762205997632838444, 1.46433900098329622080759239298, 1.59463382226284353460502792814, 2.33229638549031083540767781806, 2.65213095962528783073089481806, 2.87427635825839520143647599459, 3.27113235036049976245479313914, 3.57268462936743338421946780884, 3.72519515941886934074736349436, 4.53518390951766799385481065313, 4.63193146403438664272575340892, 5.09891356320569881797998139412, 5.43998430846459827254227490325, 5.49563093099753924726975872492, 5.71209538941045048665919269835, 6.39369944272688583628923700509, 6.78332281927936503136572651972, 6.88389736978924424688416294339, 7.20358123533573422575507075440, 7.43698690249721245608208939815, 7.66705103341604041720063058618, 7.74625580802221601183162688047, 8.141125406824900167206485656212, 8.315199211438311139449602733813
