L(s) = 1 | − 2.12·2-s + 2.50·4-s + 5-s + 1.46·7-s − 1.07·8-s − 2.12·10-s + 0.344·11-s − 3.11·14-s − 2.73·16-s + 5.13·17-s + 0.506·19-s + 2.50·20-s − 0.730·22-s + 1.07·23-s + 25-s + 3.67·28-s + 8.04·29-s + 9.17·31-s + 7.94·32-s − 10.9·34-s + 1.46·35-s + 2.73·37-s − 1.07·38-s − 1.07·40-s + 6.12·41-s − 5.87·43-s + 0.862·44-s + ⋯ |
L(s) = 1 | − 1.50·2-s + 1.25·4-s + 0.447·5-s + 0.554·7-s − 0.380·8-s − 0.671·10-s + 0.103·11-s − 0.832·14-s − 0.682·16-s + 1.24·17-s + 0.116·19-s + 0.560·20-s − 0.155·22-s + 0.224·23-s + 0.200·25-s + 0.694·28-s + 1.49·29-s + 1.64·31-s + 1.40·32-s − 1.86·34-s + 0.247·35-s + 0.448·37-s − 0.174·38-s − 0.169·40-s + 0.956·41-s − 0.895·43-s + 0.130·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.308261485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308261485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 0.344T + 11T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 - 0.506T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 - 9.17T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 6.12T + 41T^{2} \) |
| 43 | \( 1 + 5.87T + 43T^{2} \) |
| 47 | \( 1 - 4.26T + 47T^{2} \) |
| 53 | \( 1 + 4.07T + 53T^{2} \) |
| 59 | \( 1 + 5.06T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 - 0.445T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 0.834T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−7.924954558619463253264375698355, −7.57348908687628670744837960802, −6.57885218934201123001748072847, −6.12435004798216208020057629812, −5.03340801056078967260703128706, −4.49331261256861775289295550525, −3.20375829656952856789392041089, −2.40873308154956628981895829592, −1.40634333539650549797503295885, −0.815166266494946362532299838306,
0.815166266494946362532299838306, 1.40634333539650549797503295885, 2.40873308154956628981895829592, 3.20375829656952856789392041089, 4.49331261256861775289295550525, 5.03340801056078967260703128706, 6.12435004798216208020057629812, 6.57885218934201123001748072847, 7.57348908687628670744837960802, 7.924954558619463253264375698355