L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.707 + 1.22i)3-s + (2.20 + 3.82i)5-s + 2·6-s − 2.82·8-s + (0.500 + 0.866i)9-s + (3.12 − 5.40i)10-s + (2.12 − 3.67i)11-s + 13-s − 6.24·15-s + (2.00 + 3.46i)16-s + (−0.707 + 1.22i)17-s + (0.707 − 1.22i)18-s + (0.621 + 1.07i)19-s − 6·22-s + (0.0857 + 0.148i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)2-s + (−0.408 + 0.707i)3-s + (0.987 + 1.70i)5-s + 0.816·6-s − 0.999·8-s + (0.166 + 0.288i)9-s + (0.987 − 1.70i)10-s + (0.639 − 1.10i)11-s + 0.277·13-s − 1.61·15-s + (0.500 + 0.866i)16-s + (−0.171 + 0.297i)17-s + (0.166 − 0.288i)18-s + (0.142 + 0.246i)19-s − 1.27·22-s + (0.0178 + 0.0309i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09082 + 0.457146i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09082 + 0.457146i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (0.707 + 1.22i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.20 - 3.82i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.707 - 1.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.621 - 1.07i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0857 - 0.148i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 + (2.62 - 4.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.12 - 5.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (-2.20 - 3.82i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.91 + 5.04i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.82 + 10.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.24 - 2.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + (0.378 - 0.655i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.742 - 1.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.75T + 83T^{2} \) |
| 89 | \( 1 + (-2.20 - 3.82i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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
−10.59267327462456019297914165786, −10.12529404094394393722676073262, −9.468500685116332198875388451145, −8.388054381107170708078748208662, −6.83984211828681445179455554497, −6.20515507935918723280060331541, −5.40580891908357875214208548030, −3.66279227501825692805295072369, −2.84923327012054023522828296626, −1.64140424102479554315539316918,
0.838412018060798833943249554170, 2.07499416654213786082039440778, 4.19275919372652366155879826934, 5.31391616402217515650348523821, 6.15495309747039802844575875501, 6.86330930259208997777013548928, 7.77601744910836280493156015121, 8.794412358520122750607262503700, 9.287578253601421903214025818765, 10.00913047054711464455404736316