| L(s) = 1 | + 5·5-s − 5.78·7-s + 11.2·11-s + 39.4·13-s + 19.6·17-s − 102.·19-s − 175.·23-s + 25·25-s − 64.6·29-s + 193.·31-s − 28.9·35-s − 261.·37-s − 207.·41-s + 41.1·43-s + 577.·47-s − 309.·49-s − 322.·53-s + 56.2·55-s + 188.·59-s + 225.·61-s + 197.·65-s + 419.·67-s − 361.·71-s − 779.·73-s − 65.0·77-s + 696.·79-s + 333.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.312·7-s + 0.308·11-s + 0.841·13-s + 0.280·17-s − 1.23·19-s − 1.59·23-s + 0.200·25-s − 0.413·29-s + 1.12·31-s − 0.139·35-s − 1.16·37-s − 0.791·41-s + 0.146·43-s + 1.79·47-s − 0.902·49-s − 0.834·53-s + 0.137·55-s + 0.415·59-s + 0.472·61-s + 0.376·65-s + 0.765·67-s − 0.603·71-s − 1.24·73-s − 0.0963·77-s + 0.992·79-s + 0.440·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 5.78T + 343T^{2} \) |
| 11 | \( 1 - 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 39.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 64.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 207.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 41.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 322.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 188.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 225.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 419.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 361.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 779.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 696.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 333.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 570.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−8.600849100921332329137625383721, −8.013299450505623547976390685575, −6.81955057966959113710500325926, −6.23773259734104107860965988261, −5.50734290875000492212692451670, −4.32081862581314240735654327275, −3.57733946471029803133143086925, −2.37489702228133640332121696757, −1.39835829174468961031598386460, 0,
1.39835829174468961031598386460, 2.37489702228133640332121696757, 3.57733946471029803133143086925, 4.32081862581314240735654327275, 5.50734290875000492212692451670, 6.23773259734104107860965988261, 6.81955057966959113710500325926, 8.013299450505623547976390685575, 8.600849100921332329137625383721
