L(s) = 1 | + (1.98 − 4.80i)3-s − 11.9·5-s + 22.6i·7-s + (−19.1 − 19.0i)9-s − 61.9i·11-s + 71.3i·13-s + (−23.7 + 57.5i)15-s + 74.3i·17-s + 108.·19-s + (109. + 44.9i)21-s + 24.7·23-s + 18.6·25-s + (−129. + 54.2i)27-s + 84.1·29-s − 130. i·31-s + ⋯ |
L(s) = 1 | + (0.381 − 0.924i)3-s − 1.07·5-s + 1.22i·7-s + (−0.709 − 0.705i)9-s − 1.69i·11-s + 1.52i·13-s + (−0.408 + 0.990i)15-s + 1.06i·17-s + 1.30·19-s + (1.13 + 0.467i)21-s + 0.223·23-s + 0.149·25-s + (−0.922 + 0.386i)27-s + 0.538·29-s − 0.755i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.592305999\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.592305999\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.98 + 4.80i)T \) |
good | 5 | \( 1 + 11.9T + 125T^{2} \) |
| 7 | \( 1 - 22.6iT - 343T^{2} \) |
| 11 | \( 1 + 61.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 71.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 74.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 84.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 130. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 397. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 104. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 161.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 136.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 243. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 358. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 449.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 55.9iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 928. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 853. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 714.T + 9.12e5T^{2} \) |
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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
−9.319858977558989134230009555230, −8.781451783718168258234415864701, −8.122591281658027865479148257442, −7.31155866327833637968079557989, −6.22655527866209740748596818431, −5.60706867949213955582481596400, −3.98659385593583424613768162508, −3.14395516576421969730972359968, −1.98420245944748182024804946065, −0.57113392962517758285710394361,
0.856499148492855464390948390948, 2.85091596234192445406053172474, 3.65679449476118595826920459418, 4.61917462499503298646885239898, 5.15862701780866053673474088444, 6.96914291019791017436236372069, 7.62871307457615517967571799658, 8.163071845280303469890600829751, 9.529769299241894724943708233243, 10.03350102233437683075182107664