| L(s) = 1 | + 0.463·3-s + 0.183·5-s − 7-s − 2.78·9-s + 4.93·11-s + 2.19·13-s + 0.0848·15-s + 17-s − 7.89·19-s − 0.463·21-s − 3.58·23-s − 4.96·25-s − 2.68·27-s + 2.53·29-s + 1.62·31-s + 2.28·33-s − 0.183·35-s + 9.29·37-s + 1.01·39-s − 0.851·41-s + 0.271·43-s − 0.509·45-s + 10.3·47-s + 49-s + 0.463·51-s − 13.0·53-s + 0.902·55-s + ⋯ |
| L(s) = 1 | + 0.267·3-s + 0.0818·5-s − 0.377·7-s − 0.928·9-s + 1.48·11-s + 0.609·13-s + 0.0219·15-s + 0.242·17-s − 1.81·19-s − 0.101·21-s − 0.746·23-s − 0.993·25-s − 0.516·27-s + 0.471·29-s + 0.291·31-s + 0.398·33-s − 0.0309·35-s + 1.52·37-s + 0.163·39-s − 0.132·41-s + 0.0414·43-s − 0.0759·45-s + 1.50·47-s + 0.142·49-s + 0.0649·51-s − 1.78·53-s + 0.121·55-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 | \( 1 - 0.463T + 3T^{2} \) |
| 5 | \( 1 - 0.183T + 5T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 19 | \( 1 + 7.89T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 9.29T + 37T^{2} \) |
| 41 | \( 1 + 0.851T + 41T^{2} \) |
| 43 | \( 1 - 0.271T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 61 | \( 1 + 5.62T + 61T^{2} \) |
| 67 | \( 1 + 4.20T + 67T^{2} \) |
| 71 | \( 1 + 6.04T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 1.13T + 89T^{2} \) |
| 97 | \( 1 - 6.87T + 97T^{2} \) |
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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
−7.66391356034509558360029208896, −6.59969640433426702469207622280, −6.15625455589229191259511548915, −5.75571076395726601343033549671, −4.36003957850684604320387796901, −4.04081028011911900899804640386, −3.12411936873718029694890683947, −2.28852458658223760108746048702, −1.33862233878362576704371610683, 0,
1.33862233878362576704371610683, 2.28852458658223760108746048702, 3.12411936873718029694890683947, 4.04081028011911900899804640386, 4.36003957850684604320387796901, 5.75571076395726601343033549671, 6.15625455589229191259511548915, 6.59969640433426702469207622280, 7.66391356034509558360029208896
