| L(s) = 1 | − 2-s + 3-s + 4-s + 1.42·5-s − 6-s − 2.28·7-s − 8-s + 9-s − 1.42·10-s − 0.378·11-s + 12-s + 13-s + 2.28·14-s + 1.42·15-s + 16-s − 1.07·17-s − 18-s − 6.85·19-s + 1.42·20-s − 2.28·21-s + 0.378·22-s − 1.64·23-s − 24-s − 2.95·25-s − 26-s + 27-s − 2.28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.638·5-s − 0.408·6-s − 0.862·7-s − 0.353·8-s + 0.333·9-s − 0.451·10-s − 0.114·11-s + 0.288·12-s + 0.277·13-s + 0.609·14-s + 0.368·15-s + 0.250·16-s − 0.261·17-s − 0.235·18-s − 1.57·19-s + 0.319·20-s − 0.497·21-s + 0.0806·22-s − 0.343·23-s − 0.204·24-s − 0.591·25-s − 0.196·26-s + 0.192·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 1.42T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 + 0.378T + 11T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 + 5.09T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 9.76T + 37T^{2} \) |
| 41 | \( 1 - 0.952T + 41T^{2} \) |
| 43 | \( 1 - 2.65T + 43T^{2} \) |
| 47 | \( 1 - 7.38T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 + 0.731T + 59T^{2} \) |
| 61 | \( 1 + 4.05T + 61T^{2} \) |
| 67 | \( 1 + 9.03T + 67T^{2} \) |
| 71 | \( 1 - 3.02T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 5.25T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 5.26T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.64258013476262964925903762535, −6.79478153197776303077590060269, −6.18180866831586161255518738842, −5.78394048409819728155159858841, −4.45237439463310934752938976654, −3.86129004775545368488893464815, −2.72257931028873323430232871652, −2.34820819707897882151123709867, −1.29585754440563879879002959740, 0,
1.29585754440563879879002959740, 2.34820819707897882151123709867, 2.72257931028873323430232871652, 3.86129004775545368488893464815, 4.45237439463310934752938976654, 5.78394048409819728155159858841, 6.18180866831586161255518738842, 6.79478153197776303077590060269, 7.64258013476262964925903762535
