| L(s) = 1 | − 2-s + 0.986·3-s + 4-s + 2.65·5-s − 0.986·6-s − 2.59·7-s − 8-s − 2.02·9-s − 2.65·10-s + 5.12·11-s + 0.986·12-s − 4.19·13-s + 2.59·14-s + 2.62·15-s + 16-s + 6.87·17-s + 2.02·18-s − 3.27·19-s + 2.65·20-s − 2.56·21-s − 5.12·22-s − 9.24·23-s − 0.986·24-s + 2.05·25-s + 4.19·26-s − 4.95·27-s − 2.59·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.569·3-s + 0.5·4-s + 1.18·5-s − 0.402·6-s − 0.981·7-s − 0.353·8-s − 0.675·9-s − 0.840·10-s + 1.54·11-s + 0.284·12-s − 1.16·13-s + 0.693·14-s + 0.676·15-s + 0.250·16-s + 1.66·17-s + 0.477·18-s − 0.751·19-s + 0.594·20-s − 0.558·21-s − 1.09·22-s − 1.92·23-s − 0.201·24-s + 0.411·25-s + 0.823·26-s − 0.954·27-s − 0.490·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 0.986T + 3T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 9.24T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 + 0.502T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 59 | \( 1 + 1.77T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 5.57T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 0.404T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 7.57T + 89T^{2} \) |
| 97 | \( 1 - 6.10T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.136950966257357215325076813654, −7.48815294841240624012580942068, −6.52500045319980400298290309193, −6.03985049761596191073739329517, −5.43221641946297699430173327489, −3.94691424100926776825881977884, −3.24312716354948046027245700294, −2.27630875113559365022668831304, −1.61057297226619292737242463968, 0,
1.61057297226619292737242463968, 2.27630875113559365022668831304, 3.24312716354948046027245700294, 3.94691424100926776825881977884, 5.43221641946297699430173327489, 6.03985049761596191073739329517, 6.52500045319980400298290309193, 7.48815294841240624012580942068, 8.136950966257357215325076813654
