| L(s) = 1 | − 3-s − 1.07·5-s + 4.34·7-s + 9-s + 3.41·11-s − 2·13-s + 1.07·15-s − 2.34·17-s − 0.921·19-s − 4.34·21-s − 23-s − 3.83·25-s − 27-s − 6.68·29-s − 8.68·31-s − 3.41·33-s − 4.68·35-s − 7.26·37-s + 2·39-s + 8.83·41-s − 7.75·43-s − 1.07·45-s − 12.6·47-s + 11.8·49-s + 2.34·51-s + 11.9·53-s − 3.68·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.482·5-s + 1.64·7-s + 0.333·9-s + 1.03·11-s − 0.554·13-s + 0.278·15-s − 0.567·17-s − 0.211·19-s − 0.947·21-s − 0.208·23-s − 0.767·25-s − 0.192·27-s − 1.24·29-s − 1.55·31-s − 0.595·33-s − 0.791·35-s − 1.19·37-s + 0.320·39-s + 1.38·41-s − 1.18·43-s − 0.160·45-s − 1.84·47-s + 1.69·49-s + 0.327·51-s + 1.63·53-s − 0.497·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 + 0.921T + 19T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 - 8.83T + 41T^{2} \) |
| 43 | \( 1 + 7.75T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.84T + 59T^{2} \) |
| 61 | \( 1 + 4.73T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 - 2.52T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 0.894T + 83T^{2} \) |
| 89 | \( 1 + 8.49T + 89T^{2} \) |
| 97 | \( 1 - 8.15T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.892892871721117902369758717499, −7.32370958582130289412968862478, −6.63562723068034193059047497629, −5.60383115300360676625451852501, −5.06422217735745603011956784287, −4.21964348941398900189670453088, −3.70968415423723505648689573417, −2.09669515846244164470487574253, −1.48693190551744049163885876296, 0,
1.48693190551744049163885876296, 2.09669515846244164470487574253, 3.70968415423723505648689573417, 4.21964348941398900189670453088, 5.06422217735745603011956784287, 5.60383115300360676625451852501, 6.63562723068034193059047497629, 7.32370958582130289412968862478, 7.892892871721117902369758717499
