L(s) = 1 | − 1.73i·2-s − 0.999·4-s − 3.46i·7-s − 1.73i·8-s + 3.46i·11-s + (−1 + 3.46i)13-s − 5.99·14-s − 5·16-s + 6·17-s + 3.46i·19-s + 5.99·22-s + 5·25-s + (5.99 + 1.73i)26-s + 3.46i·28-s − 6·29-s + ⋯ |
L(s) = 1 | − 1.22i·2-s − 0.499·4-s − 1.30i·7-s − 0.612i·8-s + 1.04i·11-s + (−0.277 + 0.960i)13-s − 1.60·14-s − 1.25·16-s + 1.45·17-s + 0.794i·19-s + 1.27·22-s + 25-s + (1.17 + 0.339i)26-s + 0.654i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.656312 - 0.872573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.656312 - 0.872573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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
−12.94653876664632992175973565717, −12.12513433543839575821358493524, −11.16719593149364785823632598205, −10.10812231428556152192195977023, −9.615125411710301297732032956462, −7.71327862265261631414445850877, −6.72093424702019735750393022276, −4.60565980587590345138887976369, −3.47018727620373522194747060402, −1.57686584716747167281817741029,
2.91134212354398266583078098641, 5.34017448720661284081589497718, 5.82159304803056284359610985675, 7.26576752323852867591516307774, 8.340846955215312984274751734418, 9.132630457026905491932237417534, 10.74876269550658365604509198493, 11.87358826855113987605804394838, 12.92870767676208291067046516078, 14.25796646071668065415399511013