L(s) = 1 | − 2.23·2-s + 2.98·4-s + 5-s + 1.87·7-s − 2.20·8-s − 2.23·10-s − 2.02·11-s + 5.20·13-s − 4.17·14-s − 1.05·16-s + 1.73·17-s − 5.29·19-s + 2.98·20-s + 4.53·22-s − 3.33·23-s + 25-s − 11.6·26-s + 5.58·28-s + 3.53·29-s − 4.98·31-s + 6.75·32-s − 3.88·34-s + 1.87·35-s − 7.17·37-s + 11.8·38-s − 2.20·40-s + 1.08·41-s + ⋯ |
L(s) = 1 | − 1.57·2-s + 1.49·4-s + 0.447·5-s + 0.706·7-s − 0.779·8-s − 0.706·10-s − 0.611·11-s + 1.44·13-s − 1.11·14-s − 0.262·16-s + 0.421·17-s − 1.21·19-s + 0.667·20-s + 0.965·22-s − 0.695·23-s + 0.200·25-s − 2.27·26-s + 1.05·28-s + 0.657·29-s − 0.895·31-s + 1.19·32-s − 0.666·34-s + 0.316·35-s − 1.17·37-s + 1.91·38-s − 0.348·40-s + 0.169·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 - 5.20T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 + 7.17T + 37T^{2} \) |
| 41 | \( 1 - 1.08T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 + 7.45T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 3.66T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 9.52T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 - 1.11T + 73T^{2} \) |
| 79 | \( 1 + 6.37T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.399013250017887316809026624357, −7.68726710243763894327397606154, −6.75030951538665387578134252615, −6.15383888112426620499133653209, −5.23584889385749985958688611012, −4.26785720196981901954513981987, −3.10989391739364177464801798496, −1.91200569855489123789239403615, −1.42994015789105176831538915534, 0,
1.42994015789105176831538915534, 1.91200569855489123789239403615, 3.10989391739364177464801798496, 4.26785720196981901954513981987, 5.23584889385749985958688611012, 6.15383888112426620499133653209, 6.75030951538665387578134252615, 7.68726710243763894327397606154, 8.399013250017887316809026624357