L(s) = 1 | + (−2.30 − 0.617i)3-s + (1.37 + 1.76i)5-s + (−0.755 − 2.53i)7-s + (2.33 + 1.34i)9-s + (2.18 + 3.78i)11-s + (4.36 − 4.36i)13-s + (−2.09 − 4.91i)15-s + (0.438 − 1.63i)17-s + (3.56 − 6.17i)19-s + (0.175 + 6.31i)21-s + (4.91 − 1.31i)23-s + (−1.19 + 4.85i)25-s + (0.510 + 0.510i)27-s + 1.33i·29-s + (−1.90 + 1.09i)31-s + ⋯ |
L(s) = 1 | + (−1.33 − 0.356i)3-s + (0.616 + 0.787i)5-s + (−0.285 − 0.958i)7-s + (0.778 + 0.449i)9-s + (0.659 + 1.14i)11-s + (1.21 − 1.21i)13-s + (−0.539 − 1.26i)15-s + (0.106 − 0.396i)17-s + (0.818 − 1.41i)19-s + (0.0382 + 1.37i)21-s + (1.02 − 0.274i)23-s + (−0.239 + 0.970i)25-s + (0.0982 + 0.0982i)27-s + 0.247i·29-s + (−0.341 + 0.197i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.926562 - 0.242460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.926562 - 0.242460i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.37 - 1.76i)T \) |
| 7 | \( 1 + (0.755 + 2.53i)T \) |
good | 3 | \( 1 + (2.30 + 0.617i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.36 + 4.36i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.438 + 1.63i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.56 + 6.17i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.91 + 1.31i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 1.33iT - 29T^{2} \) |
| 31 | \( 1 + (1.90 - 1.09i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.224 + 0.839i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 5.69iT - 41T^{2} \) |
| 43 | \( 1 + (-3.40 - 3.40i)T + 43iT^{2} \) |
| 47 | \( 1 + (9.84 - 2.63i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.541 - 2.02i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.56 - 4.44i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.21 - 3.58i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.11 + 1.90i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 4.31T + 71T^{2} \) |
| 73 | \( 1 + (14.2 + 3.82i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.82 - 2.78i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.272 - 0.272i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.79 + 3.10i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.325 - 0.325i)T + 97iT^{2} \) |
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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
−11.54085697298527153619375307075, −10.89681298711212699148975283852, −10.23335790526656404863099010395, −9.178643430772850348459196355577, −7.31224455914434521374073288222, −6.86389219637205589876628452275, −5.90785035771100252465086841066, −4.83167336983307717177219147944, −3.20175431647358152033747381935, −1.07355853791544277786874171061,
1.38602159785301907302774112136, 3.69294591179138924691995671240, 5.10598554329358363098592199129, 5.96271742606496837298880130923, 6.36398850194175277186393501073, 8.412941873760906609315620140006, 9.135903529870612504098530144936, 10.05650818647844069945049541946, 11.35187825500389106520842654832, 11.65738370515414410476357438030