| L(s) = 1 | + 10.5·5-s + 8·7-s + 15.8·11-s − 52.9·13-s + 14·17-s + 37.0·19-s − 152·23-s − 12.9·25-s − 158.·29-s + 224·31-s + 84.6·35-s + 243.·37-s − 70·41-s − 439.·43-s − 336·47-s − 279·49-s − 31.7·53-s + 168.·55-s − 534.·59-s − 95.2·61-s − 560·65-s − 174.·67-s − 72·71-s + 294·73-s + 126.·77-s − 464·79-s − 545.·83-s + ⋯ |
| L(s) = 1 | + 0.946·5-s + 0.431·7-s + 0.435·11-s − 1.12·13-s + 0.199·17-s + 0.447·19-s − 1.37·23-s − 0.103·25-s − 1.01·29-s + 1.29·31-s + 0.408·35-s + 1.08·37-s − 0.266·41-s − 1.55·43-s − 1.04·47-s − 0.813·49-s − 0.0822·53-s + 0.411·55-s − 1.17·59-s − 0.199·61-s − 1.06·65-s − 0.318·67-s − 0.120·71-s + 0.471·73-s + 0.187·77-s − 0.660·79-s − 0.720·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 10.5T + 125T^{2} \) |
| 7 | \( 1 - 8T + 343T^{2} \) |
| 11 | \( 1 - 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152T + 1.21e4T^{2} \) |
| 29 | \( 1 + 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 + 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 336T + 1.03e5T^{2} \) |
| 53 | \( 1 + 31.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 95.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 72T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294T + 3.89e5T^{2} \) |
| 79 | \( 1 + 464T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994T + 9.12e5T^{2} \) |
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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
−8.157858570179892080433455744861, −7.61868800074260553561676845739, −6.58528589900811511408875168801, −5.94863674875556486582052429206, −5.10711409356413635113510856554, −4.37568169389757031422688125671, −3.20572995311429181443952151534, −2.16409723567548868236602726302, −1.45386576889917842104640454744, 0,
1.45386576889917842104640454744, 2.16409723567548868236602726302, 3.20572995311429181443952151534, 4.37568169389757031422688125671, 5.10711409356413635113510856554, 5.94863674875556486582052429206, 6.58528589900811511408875168801, 7.61868800074260553561676845739, 8.157858570179892080433455744861
