| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 2·10-s − 11·11-s − 12·12-s − 43·13-s − 14·14-s − 3·15-s + 16·16-s + 100·17-s + 18·18-s − 87·19-s + 4·20-s + 21·21-s − 22·22-s − 58·23-s − 24·24-s − 124·25-s − 86·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.0894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.0632·10-s − 0.301·11-s − 0.288·12-s − 0.917·13-s − 0.267·14-s − 0.0516·15-s + 1/4·16-s + 1.42·17-s + 0.235·18-s − 1.05·19-s + 0.0447·20-s + 0.218·21-s − 0.213·22-s − 0.525·23-s − 0.204·24-s − 0.991·25-s − 0.648·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 - T + p^{3} T^{2} \) |
| 13 | \( 1 + 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 100 T + p^{3} T^{2} \) |
| 19 | \( 1 + 87 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 223 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - p T + p^{3} T^{2} \) |
| 41 | \( 1 - 128 T + p^{3} T^{2} \) |
| 43 | \( 1 + 458 T + p^{3} T^{2} \) |
| 47 | \( 1 + 341 T + p^{3} T^{2} \) |
| 53 | \( 1 + 342 T + p^{3} T^{2} \) |
| 59 | \( 1 + 105 T + p^{3} T^{2} \) |
| 61 | \( 1 - 190 T + p^{3} T^{2} \) |
| 67 | \( 1 + 579 T + p^{3} T^{2} \) |
| 71 | \( 1 - 128 T + p^{3} T^{2} \) |
| 73 | \( 1 + 161 T + p^{3} T^{2} \) |
| 79 | \( 1 + 396 T + p^{3} T^{2} \) |
| 83 | \( 1 + 420 T + p^{3} T^{2} \) |
| 89 | \( 1 + 798 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1414 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18239039261953249571980998604, −9.678656207262640074296171864295, −8.107861945681130758424029173634, −7.26152122927396928105424834843, −6.18835380869119872570346214046, −5.46616914785977869841965449686, −4.42993825397833289775223072476, −3.26586092151655815076882289958, −1.87500502197503811572350213528, 0,
1.87500502197503811572350213528, 3.26586092151655815076882289958, 4.42993825397833289775223072476, 5.46616914785977869841965449686, 6.18835380869119872570346214046, 7.26152122927396928105424834843, 8.107861945681130758424029173634, 9.678656207262640074296171864295, 10.18239039261953249571980998604
