| L(s) = 1 | + 2.24·2-s − 3-s + 3.03·4-s + 3.33·5-s − 2.24·6-s + 0.336·7-s + 2.31·8-s + 9-s + 7.47·10-s + 5.35·11-s − 3.03·12-s − 2.46·13-s + 0.754·14-s − 3.33·15-s − 0.877·16-s + 3.61·17-s + 2.24·18-s − 5.53·19-s + 10.0·20-s − 0.336·21-s + 12.0·22-s + 23-s − 2.31·24-s + 6.10·25-s − 5.53·26-s − 27-s + 1.01·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.51·4-s + 1.49·5-s − 0.915·6-s + 0.127·7-s + 0.817·8-s + 0.333·9-s + 2.36·10-s + 1.61·11-s − 0.874·12-s − 0.684·13-s + 0.201·14-s − 0.860·15-s − 0.219·16-s + 0.876·17-s + 0.528·18-s − 1.27·19-s + 2.25·20-s − 0.0734·21-s + 2.56·22-s + 0.208·23-s − 0.471·24-s + 1.22·25-s − 1.08·26-s − 0.192·27-s + 0.192·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.977681527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.977681527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 7 | \( 1 - 0.336T + 7T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 + 9.57T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 3.01T + 97T^{2} \) |
| 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
−9.503505086882286802395952117990, −8.357744763587883389139682233631, −6.98258549461758788829455347271, −6.27627572583328198348609747135, −6.09699710860893165345320421522, −5.02771326023087997239369655660, −4.54265096291421056582655482382, −3.47760320058625112880368497327, −2.37037400981528006069828400248, −1.43393290452489440349337932580,
1.43393290452489440349337932580, 2.37037400981528006069828400248, 3.47760320058625112880368497327, 4.54265096291421056582655482382, 5.02771326023087997239369655660, 6.09699710860893165345320421522, 6.27627572583328198348609747135, 6.98258549461758788829455347271, 8.357744763587883389139682233631, 9.503505086882286802395952117990
