L(s) = 1 | + (−0.510 + 1.31i)2-s + 0.857·3-s + (−1.47 − 1.34i)4-s + (−1.33 − 1.79i)5-s + (−0.437 + 1.13i)6-s + (2.41 − 1.08i)7-s + (2.53 − 1.26i)8-s − 2.26·9-s + (3.04 − 0.837i)10-s + 3.05·11-s + (−1.26 − 1.15i)12-s − 3.18i·13-s + (0.206 + 3.73i)14-s + (−1.14 − 1.54i)15-s + (0.375 + 3.98i)16-s + 7.44·17-s + ⋯ |
L(s) = 1 | + (−0.360 + 0.932i)2-s + 0.495·3-s + (−0.739 − 0.673i)4-s + (−0.594 − 0.803i)5-s + (−0.178 + 0.461i)6-s + (0.911 − 0.411i)7-s + (0.894 − 0.446i)8-s − 0.754·9-s + (0.964 − 0.264i)10-s + 0.921·11-s + (−0.366 − 0.333i)12-s − 0.883i·13-s + (0.0552 + 0.998i)14-s + (−0.294 − 0.398i)15-s + (0.0939 + 0.995i)16-s + 1.80·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13447 - 0.0260320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13447 - 0.0260320i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.510 - 1.31i)T \) |
| 5 | \( 1 + (1.33 + 1.79i)T \) |
| 7 | \( 1 + (-2.41 + 1.08i)T \) |
good | 3 | \( 1 - 0.857T + 3T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 + 3.18iT - 13T^{2} \) |
| 17 | \( 1 - 7.44T + 17T^{2} \) |
| 19 | \( 1 + 4.61iT - 19T^{2} \) |
| 23 | \( 1 + 0.708T + 23T^{2} \) |
| 29 | \( 1 - 2.41iT - 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 - 5.51iT - 41T^{2} \) |
| 43 | \( 1 + 4.21iT - 43T^{2} \) |
| 47 | \( 1 - 6.55iT - 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 3.11iT - 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 + 0.519iT - 67T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 + 9.98iT - 79T^{2} \) |
| 83 | \( 1 + 8.73T + 83T^{2} \) |
| 89 | \( 1 - 18.5iT - 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.82804060382990747407074752344, −10.84960977527186755990894726688, −9.556640871960573687975708810101, −8.732530932666658726832352154195, −7.994225022662360555120252086278, −7.36280892082881350535441616758, −5.75051399304744267488541575058, −4.87595184587624073115452642728, −3.61455413123944789255666874131, −1.07218750839964494676332056105,
1.83373616902978865334386363874, 3.22388555373552400295291339950, 4.11524252622896199168668169188, 5.74131562186608453608013991276, 7.44220558639077985797081299960, 8.164187543732898409015266936977, 9.027167219204507579386355794279, 10.02852824716807371844414516388, 11.11071082379780215164278739950, 11.80739831185650835000840420749