| L(s) = 1 | + 4.09e3·2-s − 1.47e6·3-s + 1.67e7·4-s − 6.04e9·6-s + 3.57e9·7-s + 6.87e10·8-s + 1.32e12·9-s + 1.08e12·11-s − 2.47e13·12-s + 2.13e13·13-s + 1.46e13·14-s + 2.81e14·16-s − 3.79e15·17-s + 5.43e15·18-s + 6.90e15·19-s − 5.27e15·21-s + 4.44e15·22-s − 1.56e17·23-s − 1.01e17·24-s + 8.73e16·26-s − 7.08e17·27-s + 6.00e16·28-s + 6.08e17·29-s − 3.42e18·31-s + 1.15e18·32-s − 1.60e18·33-s − 1.55e19·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.60·3-s + 0.5·4-s − 1.13·6-s + 0.0977·7-s + 0.353·8-s + 1.56·9-s + 0.104·11-s − 0.801·12-s + 0.253·13-s + 0.0691·14-s + 0.250·16-s − 1.57·17-s + 1.10·18-s + 0.715·19-s − 0.156·21-s + 0.0737·22-s − 1.48·23-s − 0.566·24-s + 0.179·26-s − 0.907·27-s + 0.0488·28-s + 0.319·29-s − 0.782·31-s + 0.176·32-s − 0.167·33-s − 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.47e6T + 8.47e11T^{2} \) |
| 7 | \( 1 - 3.57e9T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.08e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 2.13e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 3.79e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 6.90e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.56e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 6.08e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.42e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 7.61e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.17e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 4.70e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 6.53e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 4.54e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.39e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 4.30e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 3.04e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 4.08e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.14e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.98e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.18e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.79e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 1.06e25T + 4.66e49T^{2} \) |
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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
−10.87574652263197860621037496963, −9.521287800440003224713119196078, −7.72811982571861579975557297708, −6.47316449232290252615907253753, −5.90102876766089973901290335234, −4.79170756460908010078615471898, −4.00693374817771153070729377968, −2.32988521467664185727467023783, −1.07253414120163122086015370079, 0,
1.07253414120163122086015370079, 2.32988521467664185727467023783, 4.00693374817771153070729377968, 4.79170756460908010078615471898, 5.90102876766089973901290335234, 6.47316449232290252615907253753, 7.72811982571861579975557297708, 9.521287800440003224713119196078, 10.87574652263197860621037496963
