| L(s) = 1 | + 2.40·2-s + 3.14·3-s + 3.76·4-s + 7.55·6-s − 2.45·7-s + 4.24·8-s + 6.90·9-s + 2.67·11-s + 11.8·12-s + 2.57·13-s − 5.89·14-s + 2.66·16-s + 16.5·18-s − 6.25·19-s − 7.72·21-s + 6.41·22-s + 3.11·23-s + 13.3·24-s + 6.17·26-s + 12.2·27-s − 9.25·28-s − 0.710·29-s + 3.99·31-s − 2.10·32-s + 8.41·33-s + 26.0·36-s + 3.63·37-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.81·3-s + 1.88·4-s + 3.08·6-s − 0.928·7-s + 1.50·8-s + 2.30·9-s + 0.805·11-s + 3.42·12-s + 0.713·13-s − 1.57·14-s + 0.665·16-s + 3.90·18-s − 1.43·19-s − 1.68·21-s + 1.36·22-s + 0.649·23-s + 2.72·24-s + 1.21·26-s + 2.36·27-s − 1.74·28-s − 0.131·29-s + 0.717·31-s − 0.371·32-s + 1.46·33-s + 4.33·36-s + 0.597·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.39583160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.39583160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 - 3.14T + 3T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 + 0.710T + 29T^{2} \) |
| 31 | \( 1 - 3.99T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 - 8.29T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 + 2.78T + 47T^{2} \) |
| 53 | \( 1 + 0.165T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 0.680T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 + 2.51T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−7.74694487302303876538474449499, −7.09769928573146526761883361347, −6.33285062890245737211510265535, −6.03016387225756184918649544798, −4.63268069759703344902140726599, −4.13306842671984761516421886585, −3.63340902935752421504515344274, −2.88591389525339884568241606753, −2.43189035274098933843103968748, −1.38118756944845780498765009545,
1.38118756944845780498765009545, 2.43189035274098933843103968748, 2.88591389525339884568241606753, 3.63340902935752421504515344274, 4.13306842671984761516421886585, 4.63268069759703344902140726599, 6.03016387225756184918649544798, 6.33285062890245737211510265535, 7.09769928573146526761883361347, 7.74694487302303876538474449499
