| L(s) = 1 | + 3.55·2-s + 4.63·4-s − 7.97·5-s − 13.2·7-s − 11.9·8-s − 28.3·10-s + 56.7·11-s + 40.1·13-s − 47.0·14-s − 79.5·16-s − 97.4·17-s + 65.2·19-s − 36.9·20-s + 201.·22-s + 124.·23-s − 61.3·25-s + 142.·26-s − 61.3·28-s + 213.·29-s − 29.5·31-s − 187.·32-s − 346.·34-s + 105.·35-s + 94.6·37-s + 232.·38-s + 95.4·40-s − 138.·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.579·4-s − 0.713·5-s − 0.714·7-s − 0.528·8-s − 0.896·10-s + 1.55·11-s + 0.856·13-s − 0.898·14-s − 1.24·16-s − 1.38·17-s + 0.788·19-s − 0.413·20-s + 1.95·22-s + 1.13·23-s − 0.491·25-s + 1.07·26-s − 0.413·28-s + 1.36·29-s − 0.171·31-s − 1.03·32-s − 1.74·34-s + 0.509·35-s + 0.420·37-s + 0.990·38-s + 0.377·40-s − 0.527·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 3.55T + 8T^{2} \) |
| 5 | \( 1 + 7.97T + 125T^{2} \) |
| 7 | \( 1 + 13.2T + 343T^{2} \) |
| 11 | \( 1 - 56.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 29.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 512.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 549.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 641.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 119.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 20.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 362.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 134.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 245.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 808.T + 9.12e5T^{2} \) |
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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
−8.456985083863016518603740623671, −7.22465632388484870876951101005, −6.44207553972007651615773017062, −6.13998541654135534077850026392, −4.84619972340328444628322769785, −4.24468521528623548805733324269, −3.51737029005153596282429729635, −2.88951562903041447427351355671, −1.32946238995772187380176392401, 0,
1.32946238995772187380176392401, 2.88951562903041447427351355671, 3.51737029005153596282429729635, 4.24468521528623548805733324269, 4.84619972340328444628322769785, 6.13998541654135534077850026392, 6.44207553972007651615773017062, 7.22465632388484870876951101005, 8.456985083863016518603740623671
