| L(s) = 1 | − 8.52·3-s + 9.40·5-s − 4.13·7-s + 45.7·9-s − 2.00·11-s − 58.7·13-s − 80.1·15-s + 83.4·19-s + 35.2·21-s + 39.2·23-s − 36.5·25-s − 159.·27-s + 157.·29-s + 120.·31-s + 17.0·33-s − 38.9·35-s − 337.·37-s + 501.·39-s − 271.·41-s + 239.·43-s + 430.·45-s − 150.·47-s − 325.·49-s − 726.·53-s − 18.8·55-s − 711.·57-s + 634.·59-s + ⋯ |
| L(s) = 1 | − 1.64·3-s + 0.841·5-s − 0.223·7-s + 1.69·9-s − 0.0549·11-s − 1.25·13-s − 1.38·15-s + 1.00·19-s + 0.366·21-s + 0.356·23-s − 0.292·25-s − 1.13·27-s + 1.01·29-s + 0.696·31-s + 0.0901·33-s − 0.187·35-s − 1.50·37-s + 2.05·39-s − 1.03·41-s + 0.848·43-s + 1.42·45-s − 0.466·47-s − 0.950·49-s − 1.88·53-s − 0.0462·55-s − 1.65·57-s + 1.39·59-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 8.52T + 27T^{2} \) |
| 5 | \( 1 - 9.40T + 125T^{2} \) |
| 7 | \( 1 + 4.13T + 343T^{2} \) |
| 11 | \( 1 + 2.00T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 83.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 239.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 150.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 634.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 617.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 712.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 830.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 299.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 614.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−8.171014752060666372377608822727, −7.13155869005372838410635524845, −6.62018380665091102653048408080, −5.86292706170918421494109874663, −5.10157838051290541009808375379, −4.75393565330001878069194418388, −3.30959891279150053712450557318, −2.10866139036407995049213862386, −1.01528031609628616112719262689, 0,
1.01528031609628616112719262689, 2.10866139036407995049213862386, 3.30959891279150053712450557318, 4.75393565330001878069194418388, 5.10157838051290541009808375379, 5.86292706170918421494109874663, 6.62018380665091102653048408080, 7.13155869005372838410635524845, 8.171014752060666372377608822727
