| L(s) = 1 | + 5·5-s − 8·7-s − 4·11-s − 6·13-s + 2·17-s − 16·19-s + 60·23-s + 25·25-s + 142·29-s − 176·31-s − 40·35-s − 214·37-s + 278·41-s − 68·43-s − 116·47-s − 279·49-s + 350·53-s − 20·55-s − 684·59-s − 394·61-s − 30·65-s + 108·67-s + 96·71-s − 398·73-s + 32·77-s + 136·79-s − 436·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.431·7-s − 0.109·11-s − 0.128·13-s + 0.0285·17-s − 0.193·19-s + 0.543·23-s + 1/5·25-s + 0.909·29-s − 1.01·31-s − 0.193·35-s − 0.950·37-s + 1.05·41-s − 0.241·43-s − 0.360·47-s − 0.813·49-s + 0.907·53-s − 0.0490·55-s − 1.50·59-s − 0.826·61-s − 0.0572·65-s + 0.196·67-s + 0.160·71-s − 0.638·73-s + 0.0473·77-s + 0.193·79-s − 0.576·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 16 T + p^{3} T^{2} \) |
| 23 | \( 1 - 60 T + p^{3} T^{2} \) |
| 29 | \( 1 - 142 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 278 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 116 T + p^{3} T^{2} \) |
| 53 | \( 1 - 350 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 394 T + p^{3} T^{2} \) |
| 67 | \( 1 - 108 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 398 T + p^{3} T^{2} \) |
| 79 | \( 1 - 136 T + p^{3} T^{2} \) |
| 83 | \( 1 + 436 T + p^{3} T^{2} \) |
| 89 | \( 1 - 750 T + p^{3} T^{2} \) |
| 97 | \( 1 - 82 T + p^{3} T^{2} \) |
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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
−8.914560998707932233712388363964, −7.959652385438611044154332111268, −7.05594487313866128175776368674, −6.32548120016501812629175971678, −5.46358371477024199769776913894, −4.59389700579660561897702384657, −3.46162776702748594952527898423, −2.54368728189513980580303591760, −1.37576435598610081062675045585, 0,
1.37576435598610081062675045585, 2.54368728189513980580303591760, 3.46162776702748594952527898423, 4.59389700579660561897702384657, 5.46358371477024199769776913894, 6.32548120016501812629175971678, 7.05594487313866128175776368674, 7.959652385438611044154332111268, 8.914560998707932233712388363964
