| L(s) = 1 | − 0.134·2-s − 2.58·3-s − 1.98·4-s − 1.71·5-s + 0.348·6-s + 0.536·8-s + 3.68·9-s + 0.231·10-s + 3.43·11-s + 5.12·12-s − 0.592·13-s + 4.43·15-s + 3.89·16-s + 0.811·17-s − 0.496·18-s − 1.38·19-s + 3.39·20-s − 0.463·22-s − 23-s − 1.38·24-s − 2.06·25-s + 0.0798·26-s − 1.76·27-s + 6.39·29-s − 0.597·30-s + 7.03·31-s − 1.59·32-s + ⋯ |
| L(s) = 1 | − 0.0953·2-s − 1.49·3-s − 0.990·4-s − 0.766·5-s + 0.142·6-s + 0.189·8-s + 1.22·9-s + 0.0731·10-s + 1.03·11-s + 1.47·12-s − 0.164·13-s + 1.14·15-s + 0.972·16-s + 0.196·17-s − 0.117·18-s − 0.318·19-s + 0.759·20-s − 0.0987·22-s − 0.208·23-s − 0.283·24-s − 0.412·25-s + 0.0156·26-s − 0.340·27-s + 1.18·29-s − 0.109·30-s + 1.26·31-s − 0.282·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.134T + 2T^{2} \) |
| 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + 0.592T + 13T^{2} \) |
| 17 | \( 1 - 0.811T + 17T^{2} \) |
| 19 | \( 1 + 1.38T + 19T^{2} \) |
| 29 | \( 1 - 6.39T + 29T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 7.45T + 41T^{2} \) |
| 43 | \( 1 + 8.85T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 0.998T + 59T^{2} \) |
| 61 | \( 1 + 7.86T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 0.224T + 73T^{2} \) |
| 79 | \( 1 + 7.18T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 - 8.59T + 89T^{2} \) |
| 97 | \( 1 + 8.38T + 97T^{2} \) |
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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
−9.667373652132272407503083914844, −8.498501737567128101633243800457, −7.87093153884400311122941323774, −6.62648062020478851624477710905, −6.10929169828156576902101610337, −4.87144124140411595962348644561, −4.49811659001842224459152397090, −3.42912668460487221769741071536, −1.19048916826252840745734959389, 0,
1.19048916826252840745734959389, 3.42912668460487221769741071536, 4.49811659001842224459152397090, 4.87144124140411595962348644561, 6.10929169828156576902101610337, 6.62648062020478851624477710905, 7.87093153884400311122941323774, 8.498501737567128101633243800457, 9.667373652132272407503083914844
