| L(s) = 1 | + 8·5-s − 30·7-s − 46·11-s + 92·13-s + 22·17-s − 86·19-s − 58·23-s − 25-s + 108·29-s + 136·31-s − 240·35-s − 180·37-s + 672·41-s − 70·43-s + 852·47-s + 190·49-s + 668·53-s − 368·55-s + 548·59-s + 284·61-s + 736·65-s + 1.16e3·67-s − 806·71-s − 1.51e3·73-s + 1.38e3·77-s − 22·79-s + 1.54e3·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s − 1.61·7-s − 1.26·11-s + 1.96·13-s + 0.313·17-s − 1.03·19-s − 0.525·23-s − 0.00799·25-s + 0.691·29-s + 0.787·31-s − 1.15·35-s − 0.799·37-s + 2.55·41-s − 0.248·43-s + 2.64·47-s + 0.553·49-s + 1.73·53-s − 0.902·55-s + 1.20·59-s + 0.596·61-s + 1.40·65-s + 2.11·67-s − 1.34·71-s − 2.42·73-s + 2.04·77-s − 0.0313·79-s + 2.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.627546492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627546492\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T + 13 p T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 30 T + 710 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 46 T + 1382 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 92 T + 6309 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 22 T - 2917 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 86 T + 10542 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 58 T + 24974 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 108 T + 51493 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 136 T + 12750 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 180 T + 109205 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 672 T + 249934 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 70 T + 53910 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 852 T + 388318 T^{2} - 852 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 668 T + 357854 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 548 T + 478598 T^{2} - 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 284 T + 416037 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1162 T + 914766 T^{2} - 1162 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 806 T + 876422 T^{2} + 806 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1510 T + 1282935 T^{2} + 1510 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22 T + 648318 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1540 T + 1446230 T^{2} - 1540 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1323 T + p^{3} T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 472 T + 1378542 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.26379611094222068489067293786, −10.19664236217597114654959457423, −9.538503742657354749242245282953, −8.983884790159777698015763414293, −8.739660334425426297164545648020, −8.380322030442082804821663659452, −7.62761880394120755188408956815, −7.38269044267722665315386207160, −6.54207290879840635475484073538, −6.32012786656726595919978322754, −5.93236203507000353622496411420, −5.62965643826621142352220091362, −4.96063606658122677943803049197, −4.10859532809610481243440168106, −3.80900109531899193072572742075, −3.21727699074220159934240217241, −2.41103859534800266919602528206, −2.30064946173452394325257242570, −1.02290660367226557313468749690, −0.54917508797246567512874928675,
0.54917508797246567512874928675, 1.02290660367226557313468749690, 2.30064946173452394325257242570, 2.41103859534800266919602528206, 3.21727699074220159934240217241, 3.80900109531899193072572742075, 4.10859532809610481243440168106, 4.96063606658122677943803049197, 5.62965643826621142352220091362, 5.93236203507000353622496411420, 6.32012786656726595919978322754, 6.54207290879840635475484073538, 7.38269044267722665315386207160, 7.62761880394120755188408956815, 8.380322030442082804821663659452, 8.739660334425426297164545648020, 8.983884790159777698015763414293, 9.538503742657354749242245282953, 10.19664236217597114654959457423, 10.26379611094222068489067293786
