| L(s) = 1 | + 6·5-s + 4·7-s + 36·11-s + 13·13-s − 66·17-s − 56·19-s + 96·23-s − 89·25-s − 222·29-s − 260·31-s + 24·35-s − 106·37-s + 90·41-s − 44·43-s + 168·47-s − 327·49-s − 30·53-s + 216·55-s + 348·59-s − 346·61-s + 78·65-s + 256·67-s − 168·71-s − 814·73-s + 144·77-s − 200·79-s + 1.23e3·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + 0.215·7-s + 0.986·11-s + 0.277·13-s − 0.941·17-s − 0.676·19-s + 0.870·23-s − 0.711·25-s − 1.42·29-s − 1.50·31-s + 0.115·35-s − 0.470·37-s + 0.342·41-s − 0.156·43-s + 0.521·47-s − 0.953·49-s − 0.0777·53-s + 0.529·55-s + 0.767·59-s − 0.726·61-s + 0.148·65-s + 0.466·67-s − 0.280·71-s − 1.30·73-s + 0.213·77-s − 0.284·79-s + 1.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 222 T + p^{3} T^{2} \) |
| 31 | \( 1 + 260 T + p^{3} T^{2} \) |
| 37 | \( 1 + 106 T + p^{3} T^{2} \) |
| 41 | \( 1 - 90 T + p^{3} T^{2} \) |
| 43 | \( 1 + 44 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 30 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 346 T + p^{3} T^{2} \) |
| 67 | \( 1 - 256 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 814 T + p^{3} T^{2} \) |
| 79 | \( 1 + 200 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1236 T + p^{3} T^{2} \) |
| 89 | \( 1 + 318 T + p^{3} T^{2} \) |
| 97 | \( 1 + 502 T + p^{3} T^{2} \) |
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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
−8.760655461053525863548133306562, −7.61777654038330747002093782203, −6.84475166749525088433853387170, −6.10401743314583280430330898378, −5.31912031151596970082541397253, −4.28206935869725760439513685793, −3.52955734910080000120246756361, −2.19120821516584182377322767414, −1.46632696507118953976319481681, 0,
1.46632696507118953976319481681, 2.19120821516584182377322767414, 3.52955734910080000120246756361, 4.28206935869725760439513685793, 5.31912031151596970082541397253, 6.10401743314583280430330898378, 6.84475166749525088433853387170, 7.61777654038330747002093782203, 8.760655461053525863548133306562
