L(s) = 1 | − 0.861·2-s − 2.77·3-s − 1.25·4-s + 5-s + 2.39·6-s + 0.623·7-s + 2.80·8-s + 4.70·9-s − 0.861·10-s − 0.0530·11-s + 3.49·12-s + 2.98·13-s − 0.537·14-s − 2.77·15-s + 0.0975·16-s − 17-s − 4.05·18-s + 2.89·19-s − 1.25·20-s − 1.73·21-s + 0.0456·22-s − 0.224·23-s − 7.79·24-s + 25-s − 2.57·26-s − 4.74·27-s − 0.784·28-s + ⋯ |
L(s) = 1 | − 0.609·2-s − 1.60·3-s − 0.628·4-s + 0.447·5-s + 0.976·6-s + 0.235·7-s + 0.992·8-s + 1.56·9-s − 0.272·10-s − 0.0159·11-s + 1.00·12-s + 0.829·13-s − 0.143·14-s − 0.716·15-s + 0.0243·16-s − 0.242·17-s − 0.956·18-s + 0.664·19-s − 0.281·20-s − 0.377·21-s + 0.00973·22-s − 0.0467·23-s − 1.59·24-s + 0.200·25-s − 0.505·26-s − 0.912·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 + 0.861T + 2T^{2} \) |
| 3 | \( 1 + 2.77T + 3T^{2} \) |
| 7 | \( 1 - 0.623T + 7T^{2} \) |
| 11 | \( 1 + 0.0530T + 11T^{2} \) |
| 13 | \( 1 - 2.98T + 13T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 23 | \( 1 + 0.224T + 23T^{2} \) |
| 29 | \( 1 + 1.81T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 3.50T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 - 9.54T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−7.70770863937219229608904489937, −6.91739452234132732834854998705, −6.27285941722949044801651009784, −5.49028861578325109537231972066, −5.04998600812918679407948235190, −4.32325697637698838878904752649, −3.38414138351172658728010527504, −1.76847760968465248120094899674, −1.05086099497310469694976061258, 0,
1.05086099497310469694976061258, 1.76847760968465248120094899674, 3.38414138351172658728010527504, 4.32325697637698838878904752649, 5.04998600812918679407948235190, 5.49028861578325109537231972066, 6.27285941722949044801651009784, 6.91739452234132732834854998705, 7.70770863937219229608904489937