| L(s) = 1 | + 1.91·2-s + 3-s + 1.67·4-s + 5-s + 1.91·6-s + 7-s − 0.628·8-s + 9-s + 1.91·10-s + 6.47·11-s + 1.67·12-s + 2.82·13-s + 1.91·14-s + 15-s − 4.54·16-s − 1.13·17-s + 1.91·18-s − 3.45·19-s + 1.67·20-s + 21-s + 12.3·22-s − 23-s − 0.628·24-s + 25-s + 5.40·26-s + 27-s + 1.67·28-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.577·3-s + 0.836·4-s + 0.447·5-s + 0.782·6-s + 0.377·7-s − 0.222·8-s + 0.333·9-s + 0.605·10-s + 1.95·11-s + 0.482·12-s + 0.782·13-s + 0.512·14-s + 0.258·15-s − 1.13·16-s − 0.274·17-s + 0.451·18-s − 0.791·19-s + 0.373·20-s + 0.218·21-s + 2.64·22-s − 0.208·23-s − 0.128·24-s + 0.200·25-s + 1.05·26-s + 0.192·27-s + 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.447669090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.447669090\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.91T + 2T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 29 | \( 1 - 7.31T + 29T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 4.35T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 + 0.0505T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 1.78T + 67T^{2} \) |
| 71 | \( 1 + 0.248T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 9.76T + 79T^{2} \) |
| 83 | \( 1 - 9.68T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.972357033172499193782050438771, −8.358423880483486237393002414498, −7.03941199897955743902276983643, −6.46845024148319620904504307774, −5.83382133056382188864225329274, −4.80434092928848603300564598674, −4.02076363670506975257415192206, −3.55879368643705065860750042555, −2.37191340298489634790288709821, −1.40631820763857107363054575003,
1.40631820763857107363054575003, 2.37191340298489634790288709821, 3.55879368643705065860750042555, 4.02076363670506975257415192206, 4.80434092928848603300564598674, 5.83382133056382188864225329274, 6.46845024148319620904504307774, 7.03941199897955743902276983643, 8.358423880483486237393002414498, 8.972357033172499193782050438771
