| L(s) = 1 | − 5·3-s − 5·5-s − 30·7-s − 27·9-s − 71·11-s − 35·13-s + 25·15-s − 38·19-s + 150·21-s − 5·23-s − 25·25-s + 260·27-s + 155·29-s − 88·31-s + 355·33-s + 150·35-s + 380·37-s + 175·39-s − 142·41-s + 155·43-s + 135·45-s − 455·47-s + 121·49-s − 275·53-s + 355·55-s + 190·57-s − 873·59-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.447·5-s − 1.61·7-s − 9-s − 1.94·11-s − 0.746·13-s + 0.430·15-s − 0.458·19-s + 1.55·21-s − 0.0453·23-s − 1/5·25-s + 1.85·27-s + 0.992·29-s − 0.509·31-s + 1.87·33-s + 0.724·35-s + 1.68·37-s + 0.718·39-s − 0.540·41-s + 0.549·43-s + 0.447·45-s − 1.41·47-s + 0.352·49-s − 0.712·53-s + 0.870·55-s + 0.441·57-s − 1.92·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 5 T + 52 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 2 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 30 T + 779 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 71 T + 3716 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 35 T + 3702 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9529 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T - 4378 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 155 T + 19928 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 88 T + 8718 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 380 T + 136878 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 142 T + 135458 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 155 T + 52086 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 455 T + 255038 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 275 T + 191846 T^{2} + 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 873 T + 591184 T^{2} + 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 445 T + 250812 T^{2} - 445 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 645 T + 674834 T^{2} - 645 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1712 T + 1445258 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1274 T + 1391022 T^{2} + 1274 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 90 T + 1038382 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 888 T + 1554274 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 710 T + 220818 T^{2} - 710 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.49788503722776660302967981564, −12.99159099534972308548603158684, −12.75174266183647683931669042549, −12.11275575634791519633455987561, −11.48177925080151079000655296282, −11.10944874209083542019704076214, −10.24642379119172767704449692881, −10.13738636923699982955297322701, −9.289819188237190567974266006637, −8.511644614083163592386551975397, −7.896494004176568198328012953839, −7.24869569877946835792619065648, −6.30155090606266999785009557310, −6.00316224674392827034502495593, −5.21601495993563621097175110718, −4.52575892854559574122108446838, −3.07106717455804559315000004384, −2.74949077236954508699012559256, 0, 0,
2.74949077236954508699012559256, 3.07106717455804559315000004384, 4.52575892854559574122108446838, 5.21601495993563621097175110718, 6.00316224674392827034502495593, 6.30155090606266999785009557310, 7.24869569877946835792619065648, 7.896494004176568198328012953839, 8.511644614083163592386551975397, 9.289819188237190567974266006637, 10.13738636923699982955297322701, 10.24642379119172767704449692881, 11.10944874209083542019704076214, 11.48177925080151079000655296282, 12.11275575634791519633455987561, 12.75174266183647683931669042549, 12.99159099534972308548603158684, 13.49788503722776660302967981564
