L(s) = 1 | − 2.39·2-s − 3-s + 3.74·4-s + 5-s + 2.39·6-s + 1.37·7-s − 4.19·8-s − 2·9-s − 2.39·10-s + 2.27·11-s − 3.74·12-s + 0.851·13-s − 3.29·14-s − 15-s + 2.56·16-s − 4.08·17-s + 4.79·18-s + 2.70·19-s + 3.74·20-s − 1.37·21-s − 5.45·22-s + 4.19·24-s + 25-s − 2.04·26-s + 5·27-s + 5.14·28-s + 6.68·29-s + ⋯ |
L(s) = 1 | − 1.69·2-s − 0.577·3-s + 1.87·4-s + 0.447·5-s + 0.978·6-s + 0.518·7-s − 1.48·8-s − 0.666·9-s − 0.758·10-s + 0.686·11-s − 1.08·12-s + 0.236·13-s − 0.879·14-s − 0.258·15-s + 0.640·16-s − 0.990·17-s + 1.13·18-s + 0.619·19-s + 0.838·20-s − 0.299·21-s − 1.16·22-s + 0.856·24-s + 0.200·25-s − 0.400·26-s + 0.962·27-s + 0.972·28-s + 1.24·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 0.851T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 + 7.02T + 43T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 + 6.83T + 53T^{2} \) |
| 59 | \( 1 - 2.71T + 59T^{2} \) |
| 61 | \( 1 + 0.908T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.555666859791444363441750421126, −8.030872186122644156158454312275, −6.89438377228185173451007862919, −6.57514516123195493580596243933, −5.58258803777558198374606160889, −4.74443852458376683531912040260, −3.30715922584878942316076402320, −2.10499490611731168340559204332, −1.28777096405515661583793180290, 0,
1.28777096405515661583793180290, 2.10499490611731168340559204332, 3.30715922584878942316076402320, 4.74443852458376683531912040260, 5.58258803777558198374606160889, 6.57514516123195493580596243933, 6.89438377228185173451007862919, 8.030872186122644156158454312275, 8.555666859791444363441750421126