L(s) = 1 | − 2.76·2-s − 1.75·3-s + 5.63·4-s − 3.34·5-s + 4.83·6-s − 0.176·7-s − 10.0·8-s + 0.0666·9-s + 9.25·10-s − 3.59·11-s − 9.87·12-s − 2.30·13-s + 0.487·14-s + 5.86·15-s + 16.4·16-s − 6.59·17-s − 0.184·18-s − 1.28·19-s − 18.8·20-s + 0.308·21-s + 9.92·22-s − 0.795·23-s + 17.5·24-s + 6.21·25-s + 6.36·26-s + 5.13·27-s − 0.994·28-s + ⋯ |
L(s) = 1 | − 1.95·2-s − 1.01·3-s + 2.81·4-s − 1.49·5-s + 1.97·6-s − 0.0666·7-s − 3.55·8-s + 0.0222·9-s + 2.92·10-s − 1.08·11-s − 2.84·12-s − 0.639·13-s + 0.130·14-s + 1.51·15-s + 4.12·16-s − 1.59·17-s − 0.0433·18-s − 0.294·19-s − 4.22·20-s + 0.0674·21-s + 2.11·22-s − 0.165·23-s + 3.59·24-s + 1.24·25-s + 1.24·26-s + 0.988·27-s − 0.187·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 + 0.176T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 + 6.59T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + 0.795T + 23T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 - 5.13T + 59T^{2} \) |
| 61 | \( 1 + 0.373T + 61T^{2} \) |
| 67 | \( 1 - 5.08T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 0.467T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 + 3.26T + 97T^{2} \) |
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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
−8.176492284163253023650853395823, −7.50950649770580304337702090864, −6.88477202385376340024289586344, −6.27577652106997140886644989720, −5.28564448767457634542597190826, −4.29859089429621871695717073224, −2.98813759689960149334111648204, −2.22559709171350695499106888735, −0.63451977981377217105337952481, 0,
0.63451977981377217105337952481, 2.22559709171350695499106888735, 2.98813759689960149334111648204, 4.29859089429621871695717073224, 5.28564448767457634542597190826, 6.27577652106997140886644989720, 6.88477202385376340024289586344, 7.50950649770580304337702090864, 8.176492284163253023650853395823