L(s) = 1 | + 2.25·2-s − 2.73·3-s + 3.07·4-s − 6.15·6-s − 7-s + 2.42·8-s + 4.45·9-s + 4.08·11-s − 8.40·12-s − 4.71·13-s − 2.25·14-s − 0.684·16-s − 4.75·17-s + 10.0·18-s + 7.76·19-s + 2.73·21-s + 9.21·22-s + 23-s − 6.62·24-s − 10.6·26-s − 3.98·27-s − 3.07·28-s + 4.28·29-s − 7.23·31-s − 6.39·32-s − 11.1·33-s − 10.7·34-s + ⋯ |
L(s) = 1 | + 1.59·2-s − 1.57·3-s + 1.53·4-s − 2.51·6-s − 0.377·7-s + 0.858·8-s + 1.48·9-s + 1.23·11-s − 2.42·12-s − 1.30·13-s − 0.602·14-s − 0.171·16-s − 1.15·17-s + 2.36·18-s + 1.78·19-s + 0.595·21-s + 1.96·22-s + 0.208·23-s − 1.35·24-s − 2.08·26-s − 0.766·27-s − 0.581·28-s + 0.795·29-s − 1.29·31-s − 1.13·32-s − 1.94·33-s − 1.83·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 29 | \( 1 - 4.28T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 1.52T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 + 0.417T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 2.63T + 67T^{2} \) |
| 71 | \( 1 + 4.00T + 71T^{2} \) |
| 73 | \( 1 - 4.80T + 73T^{2} \) |
| 79 | \( 1 - 7.50T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.44581413818836319920114212604, −6.90697732837106955268222185496, −6.42875108504562070977982091224, −5.65891103851321834278122999896, −5.08337748918676004882038356136, −4.53813228547285896130611877403, −3.72237785085428046679292091696, −2.78325632037478540978117781976, −1.50059416709234073229648251002, 0,
1.50059416709234073229648251002, 2.78325632037478540978117781976, 3.72237785085428046679292091696, 4.53813228547285896130611877403, 5.08337748918676004882038356136, 5.65891103851321834278122999896, 6.42875108504562070977982091224, 6.90697732837106955268222185496, 7.44581413818836319920114212604