L(s) = 1 | + 0.370i·2-s + i·3-s + 1.86·4-s + i·5-s − 0.370·6-s + (−2.39 − 1.12i)7-s + 1.43i·8-s − 9-s − 0.370·10-s + (2.11 + 2.55i)11-s + 1.86i·12-s + 5.18·13-s + (0.415 − 0.888i)14-s − 15-s + 3.19·16-s − 2.13·17-s + ⋯ |
L(s) = 1 | + 0.262i·2-s + 0.577i·3-s + 0.931·4-s + 0.447i·5-s − 0.151·6-s + (−0.905 − 0.424i)7-s + 0.506i·8-s − 0.333·9-s − 0.117·10-s + (0.636 + 0.771i)11-s + 0.537i·12-s + 1.43·13-s + (0.111 − 0.237i)14-s − 0.258·15-s + 0.798·16-s − 0.517·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.886540485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886540485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.39 + 1.12i)T \) |
| 11 | \( 1 + (-2.11 - 2.55i)T \) |
good | 2 | \( 1 - 0.370iT - 2T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 7.91T + 23T^{2} \) |
| 29 | \( 1 - 7.81iT - 29T^{2} \) |
| 31 | \( 1 - 7.84iT - 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 7.76iT - 43T^{2} \) |
| 47 | \( 1 - 5.62iT - 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 2.14iT - 59T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 8.98iT - 79T^{2} \) |
| 83 | \( 1 - 3.96T + 83T^{2} \) |
| 89 | \( 1 + 8.45iT - 89T^{2} \) |
| 97 | \( 1 - 8.31iT - 97T^{2} \) |
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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
−10.31458315240585291138632081118, −9.086227439353545226804284925136, −8.602100126301640255020244597317, −7.14631403715421361236627713507, −6.77185051378276802113244013029, −6.10980490959181517637123138397, −4.90354977773420096599118551390, −3.68660325149551209870856017677, −3.05869747214148363002998582845, −1.61529706663238222766826654112,
0.832917158078634443328141826137, 2.07618517625025349177234064479, 3.13939097706658778601633729234, 4.02727811489758080655763949044, 5.67510141452695928410288019723, 6.36289475785632937882111471055, 6.73190226367269584814789846483, 8.013112483448105982634343309566, 8.742138338254493922885773290756, 9.428780304382309771817622480769