| L(s) = 1 | − 128·4-s + 2.36e3·9-s − 6.84e3·11-s + 1.22e4·16-s + 8.67e4·19-s − 3.19e5·29-s − 2.87e5·31-s − 3.02e5·36-s + 1.29e5·41-s + 8.75e5·44-s − 2.35e5·49-s + 8.36e6·59-s − 5.61e5·61-s − 1.04e6·64-s − 1.23e6·71-s − 1.10e7·76-s − 9.31e6·79-s + 4.25e6·81-s + 3.44e7·89-s − 1.61e7·99-s + 2.98e7·101-s + 3.26e6·109-s + 4.08e7·116-s + 1.38e7·121-s + 3.67e7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s + 1.07·9-s − 1.54·11-s + 3/4·16-s + 2.90·19-s − 2.42·29-s − 1.73·31-s − 1.07·36-s + 0.293·41-s + 1.54·44-s − 2/7·49-s + 5.30·59-s − 0.316·61-s − 1/2·64-s − 0.410·71-s − 2.90·76-s − 2.12·79-s + 0.890·81-s + 5.18·89-s − 1.67·99-s + 2.88·101-s + 0.241·109-s + 2.42·116-s + 0.708·121-s + 1.73·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9594844788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9594844788\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2360 T^{2} + 145582 p^{2} T^{4} - 2360 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 3420 T + 10638598 T^{2} + 3420 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 89857568 T^{2} + 7014241465753038 T^{4} - 89857568 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 869409500 T^{2} + 502117989090826758 T^{4} - 869409500 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2282 p T + 2141455038 T^{2} - 2282 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 9325700444 T^{2} + 43916645758391783718 T^{4} - 9325700444 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 159576 T + 29581073878 T^{2} + 159576 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 143612 T + 59486595774 T^{2} + 143612 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8892054580 T^{2} + \)\(50\!\cdots\!78\)\( T^{4} + 8892054580 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 64848 T + 315504978094 T^{2} - 64848 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 80625225412 T^{2} + \)\(14\!\cdots\!18\)\( T^{4} + 80625225412 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 710636819372 T^{2} + \)\(49\!\cdots\!98\)\( T^{4} - 710636819372 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4519174746668 T^{2} + \)\(78\!\cdots\!38\)\( T^{4} - 4519174746668 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 4183662 T + 9350646993118 T^{2} - 4183662 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 280658 T + 5246022775002 T^{2} + 280658 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 3601365648140 T^{2} + \)\(34\!\cdots\!58\)\( T^{4} - 3601365648140 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 619272 T + 17339691732334 T^{2} + 619272 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 35641006357244 T^{2} + \)\(55\!\cdots\!18\)\( T^{4} - 35641006357244 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4656616 T + 21455665606878 T^{2} + 4656616 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 36000921688280 T^{2} + \)\(17\!\cdots\!58\)\( T^{4} - 36000921688280 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 17241420 T + 151950390368758 T^{2} - 17241420 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316949202504476 T^{2} + \)\(38\!\cdots\!38\)\( T^{4} - 316949202504476 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−7.28990550815471675111130636924, −6.86642510956835140230542189565, −6.85225180495391261084349143660, −6.26454687261577764805507466828, −5.73727531367993167352104024805, −5.72041538845627698507587776752, −5.66451124830493699336846978894, −5.12926881379460697788961196773, −5.06067571260904578018769038477, −4.98886373873085501703302079832, −4.55333606818626492204331741593, −3.99587276073056189473410630565, −3.92993184777026085321633149613, −3.77766104642779353112039991818, −3.28341392997561401591066391045, −3.05909228145754774955330947229, −3.05665251147027926361099582660, −2.13911201713403922531916976302, −2.08703337653047410890911649200, −1.98635549685718801247698588633, −1.49411365122902785802485719601, −0.881074178816987679875331286846, −0.78267704006292067956324551219, −0.69963189020651475621796446498, −0.11133001450165569977533997303,
0.11133001450165569977533997303, 0.69963189020651475621796446498, 0.78267704006292067956324551219, 0.881074178816987679875331286846, 1.49411365122902785802485719601, 1.98635549685718801247698588633, 2.08703337653047410890911649200, 2.13911201713403922531916976302, 3.05665251147027926361099582660, 3.05909228145754774955330947229, 3.28341392997561401591066391045, 3.77766104642779353112039991818, 3.92993184777026085321633149613, 3.99587276073056189473410630565, 4.55333606818626492204331741593, 4.98886373873085501703302079832, 5.06067571260904578018769038477, 5.12926881379460697788961196773, 5.66451124830493699336846978894, 5.72041538845627698507587776752, 5.73727531367993167352104024805, 6.26454687261577764805507466828, 6.85225180495391261084349143660, 6.86642510956835140230542189565, 7.28990550815471675111130636924
