| L(s) = 1 | + 2.34·2-s − 3-s + 3.48·4-s + 0.853·5-s − 2.34·6-s − 3.83·7-s + 3.48·8-s + 9-s + 2·10-s − 3.34·11-s − 3.48·12-s + 3.14·13-s − 8.97·14-s − 0.853·15-s + 1.19·16-s + 3.63·17-s + 2.34·18-s + 1.14·19-s + 2.97·20-s + 3.83·21-s − 7.83·22-s − 2.85·23-s − 3.48·24-s − 4.27·25-s + 7.37·26-s − 27-s − 13.3·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s − 0.577·3-s + 1.74·4-s + 0.381·5-s − 0.956·6-s − 1.44·7-s + 1.23·8-s + 0.333·9-s + 0.632·10-s − 1.00·11-s − 1.00·12-s + 0.872·13-s − 2.39·14-s − 0.220·15-s + 0.299·16-s + 0.881·17-s + 0.552·18-s + 0.262·19-s + 0.666·20-s + 0.836·21-s − 1.66·22-s − 0.595·23-s − 0.712·24-s − 0.854·25-s + 1.44·26-s − 0.192·27-s − 2.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.932022400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.932022400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 5 | \( 1 - 0.853T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 - 9.86T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 8.32T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 3.60T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
| 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
−13.28586640785591353784897274749, −12.71946295493445234269437623454, −11.76987345250675848387574567697, −10.59721445842523993615613944225, −9.567689349046625213550422192749, −7.50055697363095975182818161182, −6.02978154325852985056132767642, −5.79680387714073983696096153194, −4.12900087789777686630288841370, −2.85491513327109156080282679625,
2.85491513327109156080282679625, 4.12900087789777686630288841370, 5.79680387714073983696096153194, 6.02978154325852985056132767642, 7.50055697363095975182818161182, 9.567689349046625213550422192749, 10.59721445842523993615613944225, 11.76987345250675848387574567697, 12.71946295493445234269437623454, 13.28586640785591353784897274749
