| L(s) = 1 | + 2.23·2-s − 3-s + 3.00·4-s − 2.23·6-s + 2.23·8-s + 9-s + 6.47·11-s − 3.00·12-s + 4.47·13-s − 0.999·16-s − 2·17-s + 2.23·18-s + 2.47·19-s + 14.4·22-s − 4·23-s − 2.23·24-s + 10.0·26-s − 27-s − 2·29-s − 1.52·31-s − 6.70·32-s − 6.47·33-s − 4.47·34-s + 3.00·36-s + 6.94·37-s + 5.52·38-s − 4.47·39-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.50·4-s − 0.912·6-s + 0.790·8-s + 0.333·9-s + 1.95·11-s − 0.866·12-s + 1.24·13-s − 0.249·16-s − 0.485·17-s + 0.527·18-s + 0.567·19-s + 3.08·22-s − 0.834·23-s − 0.456·24-s + 1.96·26-s − 0.192·27-s − 0.371·29-s − 0.274·31-s − 1.18·32-s − 1.12·33-s − 0.766·34-s + 0.500·36-s + 1.14·37-s + 0.896·38-s − 0.716·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.586554898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.586554898\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.605084714092146135392632735867, −7.36242026990884799377344065406, −6.65477471878433815324520019001, −6.10608355086408323741169861026, −5.63503481568599403468685603418, −4.59342330779616719641612216256, −3.92832986459965507375876676913, −3.53301775422992055102375440473, −2.17083747928925924764881764933, −1.09965321179640228894749427548,
1.09965321179640228894749427548, 2.17083747928925924764881764933, 3.53301775422992055102375440473, 3.92832986459965507375876676913, 4.59342330779616719641612216256, 5.63503481568599403468685603418, 6.10608355086408323741169861026, 6.65477471878433815324520019001, 7.36242026990884799377344065406, 8.605084714092146135392632735867
