| L(s) = 1 | + 1.44·3-s + 1.25·5-s − 7-s − 0.918·9-s + 5.17·11-s − 2·13-s + 1.81·15-s + 17-s − 1.44·21-s − 2.88·23-s − 3.41·25-s − 5.65·27-s − 0.454·29-s − 10.9·31-s + 7.46·33-s − 1.25·35-s − 7.17·37-s − 2.88·39-s + 3.21·41-s − 9.50·43-s − 1.15·45-s − 10.1·47-s + 49-s + 1.44·51-s − 6.59·53-s + 6.51·55-s − 10.3·59-s + ⋯ |
| L(s) = 1 | + 0.832·3-s + 0.562·5-s − 0.377·7-s − 0.306·9-s + 1.56·11-s − 0.554·13-s + 0.468·15-s + 0.242·17-s − 0.314·21-s − 0.601·23-s − 0.683·25-s − 1.08·27-s − 0.0844·29-s − 1.96·31-s + 1.30·33-s − 0.212·35-s − 1.17·37-s − 0.462·39-s + 0.502·41-s − 1.44·43-s − 0.172·45-s − 1.48·47-s + 0.142·49-s + 0.202·51-s − 0.906·53-s + 0.878·55-s − 1.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.88T + 23T^{2} \) |
| 29 | \( 1 + 0.454T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 7.17T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 + 9.50T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 0.557T + 61T^{2} \) |
| 67 | \( 1 + 0.966T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 5.36T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 + 7.60T + 89T^{2} \) |
| 97 | \( 1 + 1.48T + 97T^{2} \) |
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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
−7.57870908192657600564203624104, −6.83221463365002763373471481536, −6.17796411695849308314086373784, −5.54438612605358544619140777902, −4.63101395136686085929129036214, −3.52715820691068424466392719626, −3.39652196450759112267678827434, −2.09181002117264184599933054698, −1.65178277578458985565291360045, 0,
1.65178277578458985565291360045, 2.09181002117264184599933054698, 3.39652196450759112267678827434, 3.52715820691068424466392719626, 4.63101395136686085929129036214, 5.54438612605358544619140777902, 6.17796411695849308314086373784, 6.83221463365002763373471481536, 7.57870908192657600564203624104
