L(s) = 1 | − 3.50·2-s + 8.27·4-s − 2.23i·5-s + (−6.69 − 2.03i)7-s − 14.9·8-s + 7.83i·10-s + 2.03·11-s + 18.0i·13-s + (23.4 + 7.13i)14-s + 19.3·16-s + 1.07i·17-s − 28.7i·19-s − 18.5i·20-s − 7.11·22-s − 24.8·23-s + ⋯ |
L(s) = 1 | − 1.75·2-s + 2.06·4-s − 0.447i·5-s + (−0.956 − 0.290i)7-s − 1.87·8-s + 0.783i·10-s + 0.184·11-s + 1.38i·13-s + (1.67 + 0.509i)14-s + 1.21·16-s + 0.0631i·17-s − 1.51i·19-s − 0.925i·20-s − 0.323·22-s − 1.08·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.335412 + 0.248603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.335412 + 0.248603i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (6.69 + 2.03i)T \) |
good | 2 | \( 1 + 3.50T + 4T^{2} \) |
| 11 | \( 1 - 2.03T + 121T^{2} \) |
| 13 | \( 1 - 18.0iT - 169T^{2} \) |
| 17 | \( 1 - 1.07iT - 289T^{2} \) |
| 19 | \( 1 + 28.7iT - 361T^{2} \) |
| 23 | \( 1 + 24.8T + 529T^{2} \) |
| 29 | \( 1 - 38.4T + 841T^{2} \) |
| 31 | \( 1 - 44.0iT - 961T^{2} \) |
| 37 | \( 1 - 37.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.58T + 1.84e3T^{2} \) |
| 47 | \( 1 - 55.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 57.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 95.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 25.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 95.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 28.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 29.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 67.6iT - 9.40e3T^{2} \) |
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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
−11.38357042865698302793391431214, −10.39831762279356724595718252067, −9.513482795355436961462099269987, −9.045528560180859835241532081211, −8.070972606789545075672336064249, −6.89899738089159323275156927693, −6.38331313911336425338201946039, −4.44773705314974248596570162810, −2.67819723528612444357738526283, −1.12760618796661099724134653782,
0.40519449393391429779084308196, 2.23438871562153050752757393300, 3.49630684938388322266944754967, 5.83769790258135753828314060562, 6.58541183719361465300334666477, 7.80156221412936665749525576492, 8.315219993361771782173933779051, 9.655725175409374766506454289520, 10.00558036961201462757432040366, 10.82610644694541540109863334103