| L(s) = 1 | + 4·4-s − 8·7-s − 6·9-s − 24·11-s + 12·16-s − 24·23-s − 32·28-s + 120·29-s − 24·36-s + 80·37-s + 128·43-s − 96·44-s + 22·49-s + 216·53-s + 48·63-s + 32·64-s − 176·67-s − 120·71-s + 192·77-s + 128·79-s + 27·81-s − 96·92-s + 144·99-s + 168·107-s − 8·109-s − 96·112-s + 360·113-s + ⋯ |
| L(s) = 1 | + 4-s − 8/7·7-s − 2/3·9-s − 2.18·11-s + 3/4·16-s − 1.04·23-s − 8/7·28-s + 4.13·29-s − 2/3·36-s + 2.16·37-s + 2.97·43-s − 2.18·44-s + 0.448·49-s + 4.07·53-s + 0.761·63-s + 1/2·64-s − 2.62·67-s − 1.69·71-s + 2.49·77-s + 1.62·79-s + 1/3·81-s − 1.04·92-s + 1.45·99-s + 1.57·107-s − 0.0733·109-s − 6/7·112-s + 3.18·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.378022820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.378022820\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 8 T + 6 p T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 11 | $D_{4}$ | \( ( 1 + 12 T + 260 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 53574 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 320 T^{2} + 546 p^{2} T^{4} + 320 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 652 T^{2} + 348486 T^{4} - 652 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 12 T + 932 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1252 T^{2} + 745926 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 40 T + 546 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 64 T + 2922 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3940 T^{2} + 5766 p^{2} T^{4} - 3940 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 108 T + 8246 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4420 T^{2} + 6539622 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14596 T^{2} + 80934054 T^{4} - 14596 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 88 T + 10626 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 60 T + 4484 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 13108 T^{2} + 98972646 T^{4} - 13108 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 64 T + 3138 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16612 T^{2} + 134415078 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8896 T^{2} + 52680354 T^{4} - 8896 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 24820 T^{2} + 317127462 T^{4} - 24820 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.86709731187403692778341466075, −6.70037428107849668524122431072, −6.27630536488725882706940714298, −6.20023005267359221061046968218, −6.00765617406271250456173378628, −5.96966468695974078392156136641, −5.49169445591436090978674332926, −5.46282157353359904317426169763, −5.20994963618826235376782018888, −4.80735994986352453473613060232, −4.37813821283638617929305802824, −4.36707788518781302707866997637, −4.27974133390889508857887917118, −3.78801549806953280553052930671, −3.41080952531106807111790925492, −3.04312276767932377745221787067, −2.89688097871599252226229775516, −2.72017514123958501353304404050, −2.51959211087655151597095553209, −2.32967238231995013877148911536, −2.05568446730905236888704954953, −1.39676864579915157196245261553, −0.887726375362125515536554924979, −0.60556717994091459156255885769, −0.47762188420151697561238640044,
0.47762188420151697561238640044, 0.60556717994091459156255885769, 0.887726375362125515536554924979, 1.39676864579915157196245261553, 2.05568446730905236888704954953, 2.32967238231995013877148911536, 2.51959211087655151597095553209, 2.72017514123958501353304404050, 2.89688097871599252226229775516, 3.04312276767932377745221787067, 3.41080952531106807111790925492, 3.78801549806953280553052930671, 4.27974133390889508857887917118, 4.36707788518781302707866997637, 4.37813821283638617929305802824, 4.80735994986352453473613060232, 5.20994963618826235376782018888, 5.46282157353359904317426169763, 5.49169445591436090978674332926, 5.96966468695974078392156136641, 6.00765617406271250456173378628, 6.20023005267359221061046968218, 6.27630536488725882706940714298, 6.70037428107849668524122431072, 6.86709731187403692778341466075
