| L(s) = 1 | − 2.43·2-s − 3-s + 3.92·4-s + 5-s + 2.43·6-s − 1.65·7-s − 4.68·8-s + 9-s − 2.43·10-s + 0.00347·11-s − 3.92·12-s − 4.91·13-s + 4.02·14-s − 15-s + 3.55·16-s + 0.461·17-s − 2.43·18-s + 6.59·19-s + 3.92·20-s + 1.65·21-s − 0.00844·22-s − 3.49·23-s + 4.68·24-s + 25-s + 11.9·26-s − 27-s − 6.49·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.577·3-s + 1.96·4-s + 0.447·5-s + 0.993·6-s − 0.625·7-s − 1.65·8-s + 0.333·9-s − 0.769·10-s + 0.00104·11-s − 1.13·12-s − 1.36·13-s + 1.07·14-s − 0.258·15-s + 0.888·16-s + 0.112·17-s − 0.573·18-s + 1.51·19-s + 0.877·20-s + 0.360·21-s − 0.00180·22-s − 0.729·23-s + 0.956·24-s + 0.200·25-s + 2.34·26-s − 0.192·27-s − 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4244528151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4244528151\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 - 0.00347T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 - 0.461T + 17T^{2} \) |
| 19 | \( 1 - 6.59T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 + 5.06T + 29T^{2} \) |
| 31 | \( 1 - 0.917T + 31T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 - 5.89T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 + 0.466T + 53T^{2} \) |
| 59 | \( 1 - 8.49T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 7.98T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 0.253T + 97T^{2} \) |
| 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
−8.011311998271500770724977580436, −7.36552226046127791574095810770, −7.04118941333538870886056926781, −6.10190843568841623832283135712, −5.53780606353243941742822955589, −4.58745579718169968803785615714, −3.29683142217241540036117089861, −2.41580464045397567420930571190, −1.54435928145123540464012831639, −0.46348219619525871351549268739,
0.46348219619525871351549268739, 1.54435928145123540464012831639, 2.41580464045397567420930571190, 3.29683142217241540036117089861, 4.58745579718169968803785615714, 5.53780606353243941742822955589, 6.10190843568841623832283135712, 7.04118941333538870886056926781, 7.36552226046127791574095810770, 8.011311998271500770724977580436
