| L(s) = 1 | − 3-s + 1.25·5-s − 0.924·7-s + 9-s + 4.02·11-s + 2.29·13-s − 1.25·15-s − 6.17·17-s + 1.46·19-s + 0.924·21-s − 6.97·23-s − 3.41·25-s − 27-s − 6.00·29-s − 1.16·31-s − 4.02·33-s − 1.16·35-s − 5.20·37-s − 2.29·39-s − 6.15·41-s + 3.36·43-s + 1.25·45-s + 2.22·47-s − 6.14·49-s + 6.17·51-s + 5.66·53-s + 5.06·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.562·5-s − 0.349·7-s + 0.333·9-s + 1.21·11-s + 0.635·13-s − 0.325·15-s − 1.49·17-s + 0.336·19-s + 0.201·21-s − 1.45·23-s − 0.683·25-s − 0.192·27-s − 1.11·29-s − 0.209·31-s − 0.700·33-s − 0.196·35-s − 0.855·37-s − 0.366·39-s − 0.961·41-s + 0.513·43-s + 0.187·45-s + 0.325·47-s − 0.877·49-s + 0.864·51-s + 0.778·53-s + 0.683·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 + 0.924T + 7T^{2} \) |
| 11 | \( 1 - 4.02T + 11T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 + 6.00T + 29T^{2} \) |
| 31 | \( 1 + 1.16T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 59 | \( 1 - 3.30T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 + 6.42T + 79T^{2} \) |
| 83 | \( 1 - 5.18T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 0.704T + 97T^{2} \) |
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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
−8.153693581690749349110703755963, −7.07665378574893637030490659151, −6.52999326940549642114366517539, −5.95662426933998849498651826540, −5.25740878426168749009581617706, −4.09720483572716657341926990556, −3.71387607347594524695801834098, −2.23570714394053259578544156867, −1.47976269308042810957422518239, 0,
1.47976269308042810957422518239, 2.23570714394053259578544156867, 3.71387607347594524695801834098, 4.09720483572716657341926990556, 5.25740878426168749009581617706, 5.95662426933998849498651826540, 6.52999326940549642114366517539, 7.07665378574893637030490659151, 8.153693581690749349110703755963
