| L(s) = 1 | + 10.0·3-s + 10.2·5-s − 1.06·7-s + 74.5·9-s − 23.8·11-s − 64.6·13-s + 103.·15-s + 89.6·19-s − 10.7·21-s + 203.·23-s − 19.4·25-s + 479.·27-s − 14.3·29-s + 237.·31-s − 240.·33-s − 10.9·35-s − 21.8·37-s − 651.·39-s − 342.·41-s + 452.·43-s + 765.·45-s − 61.3·47-s − 341.·49-s + 268.·53-s − 244.·55-s + 903.·57-s − 221.·59-s + ⋯ |
| L(s) = 1 | + 1.93·3-s + 0.918·5-s − 0.0576·7-s + 2.76·9-s − 0.652·11-s − 1.38·13-s + 1.78·15-s + 1.08·19-s − 0.111·21-s + 1.84·23-s − 0.155·25-s + 3.41·27-s − 0.0919·29-s + 1.37·31-s − 1.26·33-s − 0.0529·35-s − 0.0969·37-s − 2.67·39-s − 1.30·41-s + 1.60·43-s + 2.53·45-s − 0.190·47-s − 0.996·49-s + 0.696·53-s − 0.599·55-s + 2.09·57-s − 0.488·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.215454877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.215454877\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 10.0T + 27T^{2} \) |
| 5 | \( 1 - 10.2T + 125T^{2} \) |
| 7 | \( 1 + 1.06T + 343T^{2} \) |
| 11 | \( 1 + 23.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.6T + 2.19e3T^{2} \) |
| 19 | \( 1 - 89.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 203.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 14.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 21.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 452.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 61.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 268.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 221.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 69.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 922.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 414.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 910.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 27.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−8.778447744769819951581376454592, −7.86602426317240436786924636127, −7.36310127151104516454299584804, −6.62216314113742316937751714218, −5.24772144425295776177660869051, −4.66077149000127336574143700031, −3.39840120642177752008485243929, −2.72328511150086934381549067151, −2.16108859706120079436498951766, −1.06090874448195701958170736339,
1.06090874448195701958170736339, 2.16108859706120079436498951766, 2.72328511150086934381549067151, 3.39840120642177752008485243929, 4.66077149000127336574143700031, 5.24772144425295776177660869051, 6.62216314113742316937751714218, 7.36310127151104516454299584804, 7.86602426317240436786924636127, 8.778447744769819951581376454592
