| L(s) = 1 | + 0.113·2-s + 2.52·3-s − 1.98·4-s − 5-s + 0.286·6-s − 0.501·7-s − 0.451·8-s + 3.37·9-s − 0.113·10-s − 6.19·11-s − 5.01·12-s + 3.05·13-s − 0.0567·14-s − 2.52·15-s + 3.92·16-s + 6.98·17-s + 0.382·18-s − 3.54·19-s + 1.98·20-s − 1.26·21-s − 0.702·22-s − 1.14·24-s + 25-s + 0.345·26-s + 0.952·27-s + 0.996·28-s − 8.35·29-s + ⋯ |
| L(s) = 1 | + 0.0801·2-s + 1.45·3-s − 0.993·4-s − 0.447·5-s + 0.116·6-s − 0.189·7-s − 0.159·8-s + 1.12·9-s − 0.0358·10-s − 1.86·11-s − 1.44·12-s + 0.846·13-s − 0.0151·14-s − 0.652·15-s + 0.980·16-s + 1.69·17-s + 0.0901·18-s − 0.812·19-s + 0.444·20-s − 0.276·21-s − 0.149·22-s − 0.232·24-s + 0.200·25-s + 0.0678·26-s + 0.183·27-s + 0.188·28-s − 1.55·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.113T + 2T^{2} \) |
| 3 | \( 1 - 2.52T + 3T^{2} \) |
| 7 | \( 1 + 0.501T + 7T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 6.98T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 29 | \( 1 + 8.35T + 29T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 + 4.71T + 47T^{2} \) |
| 53 | \( 1 + 6.35T + 53T^{2} \) |
| 59 | \( 1 + 5.33T + 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 6.85T + 71T^{2} \) |
| 73 | \( 1 + 3.29T + 73T^{2} \) |
| 79 | \( 1 + 6.07T + 79T^{2} \) |
| 83 | \( 1 - 3.53T + 83T^{2} \) |
| 89 | \( 1 + 1.94T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.459129534854796492799456848663, −7.84521367472403831300510745714, −7.52327825452625572607679625813, −6.01614916470706322785880797749, −5.22388601574449653003191362041, −4.33560672384481287250135820782, −3.31841505379163783647820989096, −3.11256707345101862827174533409, −1.68437697414424984007838689636, 0,
1.68437697414424984007838689636, 3.11256707345101862827174533409, 3.31841505379163783647820989096, 4.33560672384481287250135820782, 5.22388601574449653003191362041, 6.01614916470706322785880797749, 7.52327825452625572607679625813, 7.84521367472403831300510745714, 8.459129534854796492799456848663
