| L(s) = 1 | + 1.96·2-s + 3-s + 1.84·4-s + 5-s + 1.96·6-s − 4.37·7-s − 0.302·8-s + 9-s + 1.96·10-s − 1.11·11-s + 1.84·12-s + 5.15·13-s − 8.58·14-s + 15-s − 4.28·16-s − 7.19·17-s + 1.96·18-s + 4.27·19-s + 1.84·20-s − 4.37·21-s − 2.19·22-s + 0.550·23-s − 0.302·24-s + 25-s + 10.1·26-s + 27-s − 8.08·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.577·3-s + 0.922·4-s + 0.447·5-s + 0.800·6-s − 1.65·7-s − 0.106·8-s + 0.333·9-s + 0.620·10-s − 0.336·11-s + 0.532·12-s + 1.42·13-s − 2.29·14-s + 0.258·15-s − 1.07·16-s − 1.74·17-s + 0.462·18-s + 0.981·19-s + 0.412·20-s − 0.955·21-s − 0.467·22-s + 0.114·23-s − 0.0617·24-s + 0.200·25-s + 1.98·26-s + 0.192·27-s − 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.96T + 2T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 + 1.11T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 - 0.550T + 23T^{2} \) |
| 29 | \( 1 + 9.72T + 29T^{2} \) |
| 31 | \( 1 + 7.42T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 + 3.97T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 + 0.0194T + 53T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 6.38T + 67T^{2} \) |
| 71 | \( 1 + 4.28T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 + 7.03T + 79T^{2} \) |
| 83 | \( 1 - 8.93T + 83T^{2} \) |
| 89 | \( 1 + 0.626T + 89T^{2} \) |
| 97 | \( 1 + 5.07T + 97T^{2} \) |
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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
−7.36791324339108819882256111316, −6.79950687150526911630216393794, −6.10109064381964947693593540477, −5.70125856787788245247682045714, −4.74866784183613036859012541059, −3.77604128667903342707700952582, −3.45877237664226875675923879798, −2.71290825298678922693846779965, −1.79253267444056814897514412120, 0,
1.79253267444056814897514412120, 2.71290825298678922693846779965, 3.45877237664226875675923879798, 3.77604128667903342707700952582, 4.74866784183613036859012541059, 5.70125856787788245247682045714, 6.10109064381964947693593540477, 6.79950687150526911630216393794, 7.36791324339108819882256111316
