| L(s) = 1 | + 2·2-s − 2·5-s − 12·7-s − 8·8-s − 4·10-s − 12·11-s − 2·13-s − 24·14-s − 16·16-s − 20·17-s − 24·22-s − 48·23-s + 25·25-s − 4·26-s − 26·29-s − 12·31-s − 40·34-s + 24·35-s + 52·37-s + 16·40-s + 58·41-s − 84·43-s − 96·46-s − 120·47-s + 47·49-s + 50·50-s + 148·53-s + ⋯ |
| L(s) = 1 | + 2-s − 2/5·5-s − 1.71·7-s − 8-s − 2/5·10-s − 1.09·11-s − 0.153·13-s − 1.71·14-s − 16-s − 1.17·17-s − 1.09·22-s − 2.08·23-s + 25-s − 0.153·26-s − 0.896·29-s − 0.387·31-s − 1.17·34-s + 0.685·35-s + 1.40·37-s + 2/5·40-s + 1.41·41-s − 1.95·43-s − 2.08·46-s − 2.55·47-s + 0.959·49-s + 50-s + 2.79·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1186909949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1186909949\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T + 169 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T - 165 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 48 T + 1297 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T - 165 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T + 1009 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 58 T + 1683 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 84 T + 4201 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 120 T + 7009 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 156 T + 11593 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T - 3045 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 4537 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 204 T + 20113 T^{2} - 204 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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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
−11.72961434489786157942847585850, −11.29387342887454806574607139596, −10.78402659816629405112350240089, −10.06030159042584875631355785039, −9.940984964279912904362777562420, −9.205668706343231434137846308469, −9.053910286263710746606062638145, −8.145003103636377732392346662195, −7.940534805112929077970162727799, −7.14482798578321806455600881390, −6.51150779221721557336402637020, −6.30318045579560529415228275301, −5.67412141302988541619372049614, −5.15759336281776685108476172514, −4.34647083206120352549128820763, −4.10245543068972086000529079829, −3.22786289856363845548582311928, −2.92711068849275110330179030511, −2.09493291240081836587441258874, −0.12925517965759771049272670754,
0.12925517965759771049272670754, 2.09493291240081836587441258874, 2.92711068849275110330179030511, 3.22786289856363845548582311928, 4.10245543068972086000529079829, 4.34647083206120352549128820763, 5.15759336281776685108476172514, 5.67412141302988541619372049614, 6.30318045579560529415228275301, 6.51150779221721557336402637020, 7.14482798578321806455600881390, 7.940534805112929077970162727799, 8.145003103636377732392346662195, 9.053910286263710746606062638145, 9.205668706343231434137846308469, 9.940984964279912904362777562420, 10.06030159042584875631355785039, 10.78402659816629405112350240089, 11.29387342887454806574607139596, 11.72961434489786157942847585850
