| L(s) = 1 | + 2.00·2-s − 2.19·3-s + 2.02·4-s + 5-s − 4.40·6-s + 0.0594·8-s + 1.80·9-s + 2.00·10-s − 0.770·11-s − 4.45·12-s + 3.71·13-s − 2.19·15-s − 3.93·16-s − 1.98·17-s + 3.62·18-s + 3.70·19-s + 2.02·20-s − 1.54·22-s − 7.32·23-s − 0.130·24-s + 25-s + 7.45·26-s + 2.61·27-s + 6.50·29-s − 4.40·30-s − 31-s − 8.02·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.26·3-s + 1.01·4-s + 0.447·5-s − 1.79·6-s + 0.0210·8-s + 0.602·9-s + 0.634·10-s − 0.232·11-s − 1.28·12-s + 1.03·13-s − 0.566·15-s − 0.984·16-s − 0.481·17-s + 0.855·18-s + 0.849·19-s + 0.453·20-s − 0.329·22-s − 1.52·23-s − 0.0266·24-s + 0.200·25-s + 1.46·26-s + 0.503·27-s + 1.20·29-s − 0.803·30-s − 0.179·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 3 | \( 1 + 2.19T + 3T^{2} \) |
| 11 | \( 1 + 0.770T + 11T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 + 1.98T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 + 1.55T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 4.45T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 - 0.787T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 7.16T + 83T^{2} \) |
| 89 | \( 1 + 1.05T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−6.96710682912937111569708591138, −6.51676230354042662476069792798, −5.93317287381985961272207983829, −5.41672670302066502503574650636, −4.90410531966716223200028490926, −4.10801233679113343766505623653, −3.38460358490655971416568361038, −2.45302342027368275242649259503, −1.35421209428974618435936490698, 0,
1.35421209428974618435936490698, 2.45302342027368275242649259503, 3.38460358490655971416568361038, 4.10801233679113343766505623653, 4.90410531966716223200028490926, 5.41672670302066502503574650636, 5.93317287381985961272207983829, 6.51676230354042662476069792798, 6.96710682912937111569708591138
