L(s) = 1 | − 1.83i·2-s + 2.40i·3-s − 1.36·4-s + (−1.46 + 1.68i)5-s + 4.41·6-s − 0.302i·7-s − 1.17i·8-s − 2.78·9-s + (3.09 + 2.69i)10-s − 5.09·11-s − 3.27i·12-s − 4.63i·13-s − 0.554·14-s + (−4.05 − 3.53i)15-s − 4.87·16-s − 1.03i·17-s + ⋯ |
L(s) = 1 | − 1.29i·2-s + 1.38i·3-s − 0.680·4-s + (−0.656 + 0.754i)5-s + 1.80·6-s − 0.114i·7-s − 0.414i·8-s − 0.929·9-s + (0.977 + 0.850i)10-s − 1.53·11-s − 0.944i·12-s − 1.28i·13-s − 0.148·14-s + (−1.04 − 0.912i)15-s − 1.21·16-s − 0.250i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7789638867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7789638867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.46 - 1.68i)T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 1.83iT - 2T^{2} \) |
| 3 | \( 1 - 2.40iT - 3T^{2} \) |
| 7 | \( 1 + 0.302iT - 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 4.63iT - 13T^{2} \) |
| 17 | \( 1 + 1.03iT - 17T^{2} \) |
| 19 | \( 1 - 8.55T + 19T^{2} \) |
| 23 | \( 1 + 1.34iT - 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.459T + 31T^{2} \) |
| 37 | \( 1 + 1.55iT - 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 + 5.45iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 4.20iT - 53T^{2} \) |
| 59 | \( 1 - 5.01T + 59T^{2} \) |
| 61 | \( 1 + 6.46T + 61T^{2} \) |
| 67 | \( 1 + 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 2.63T + 79T^{2} \) |
| 83 | \( 1 - 2.13iT - 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−9.866464851000724110158940321053, −9.101169771790301719635045382816, −7.80018649118057731614642681678, −7.28880789878092993538842445235, −5.60805060251477763575068973498, −4.94277962966098276772253882086, −3.71487045239652882991051036454, −3.30903345083082092928303447102, −2.47537626910827574215135460267, −0.33752471250844118485369141109,
1.39161902117186887677901401909, 2.66950083451573109470137651518, 4.31563684294715123306739004132, 5.39763817505646393821277099011, 5.87376097511247752454622340017, 7.11750127687440298765766915572, 7.49232853858509430394166394485, 7.928878931673530549605147608410, 8.816845977785333491987942596513, 9.608912479670940968658617810381