L(s) = 1 | − 2.64i·3-s − 3.46i·5-s − i·7-s − 4.00·9-s + 2.64·13-s − 9.16·15-s − 3·17-s + (−3.46 − 2.64i)19-s − 2.64·21-s − 3i·23-s − 6.99·25-s + 2.64i·27-s + 7.93·29-s − 3.46·35-s + 10.5·37-s + ⋯ |
L(s) = 1 | − 1.52i·3-s − 1.54i·5-s − 0.377i·7-s − 1.33·9-s + 0.733·13-s − 2.36·15-s − 0.727·17-s + (−0.794 − 0.606i)19-s − 0.577·21-s − 0.625i·23-s − 1.39·25-s + 0.509i·27-s + 1.47·29-s − 0.585·35-s + 1.73·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.376226655\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.376226655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (3.46 + 2.64i)T \) |
good | 3 | \( 1 + 2.64iT - 3T^{2} \) |
| 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 9.16iT - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 7.93T + 53T^{2} \) |
| 59 | \( 1 + 7.93iT - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 13.2iT - 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 9.16iT - 89T^{2} \) |
| 97 | \( 1 - 9.16iT - 97T^{2} \) |
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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
−8.879425601906594055442109461641, −8.399824602320880588120919805353, −7.80977523614209372230442356427, −6.62593968191993972931457919863, −6.24817789524827216156353271254, −4.95635936344124367603610832251, −4.23751512314524123948264328204, −2.58731939694500820612134034625, −1.42459055477061430825202347179, −0.61462210392373565737251189020,
2.31111772095551403053240269003, 3.30990611421250955451208655633, 3.96724887507630970068491393738, 4.97436670550036931936332119937, 6.09821268832581471594804137233, 6.61832120019357949101379547979, 7.86547163225915508941138638341, 8.750258233724518417390719603640, 9.562367964363615099913669887545, 10.29751272374416617987870902488