L(s) = 1 | − 19.8i·5-s − 19.8·7-s + 48i·11-s − 79.5i·13-s + 42·17-s − 92i·19-s − 39.7·23-s − 271·25-s − 19.8i·29-s − 139.·31-s + 396i·35-s + 198. i·37-s + 6·41-s + 92i·43-s − 39.7·47-s + ⋯ |
L(s) = 1 | − 1.77i·5-s − 1.07·7-s + 1.31i·11-s − 1.69i·13-s + 0.599·17-s − 1.11i·19-s − 0.360·23-s − 2.16·25-s − 0.127i·29-s − 0.807·31-s + 1.91i·35-s + 0.884i·37-s + 0.0228·41-s + 0.326i·43-s − 0.123·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2992180457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2992180457\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 19.8iT - 125T^{2} \) |
| 7 | \( 1 + 19.8T + 343T^{2} \) |
| 11 | \( 1 - 48iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 79.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 42T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 39.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 19.8iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 198. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 39.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 497. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 516iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 358. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 524iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 994.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 430T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 432iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 630T + 7.04e5T^{2} \) |
| 97 | \( 1 - 862T + 9.12e5T^{2} \) |
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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
−9.708870282534436276077902788277, −8.999178594735391316249340914410, −8.061942647200421762228461127057, −7.20128342874070722646190044949, −5.85855157752119952850807721658, −5.10549685591805685928974464264, −4.18019106788526052585929223640, −2.82889650211528723770220127447, −1.22067999261163554570076474870, −0.092508208861167735424008066167,
2.05584605896764347943284107815, 3.33119190085636159839446198485, 3.75734297713093941511841362275, 5.74571087730570693987680185279, 6.44322515407994678326487314230, 7.03679867617726707311363152128, 8.119702188887692381310879835778, 9.353958222036313183399609673598, 10.00355460302060978418938634548, 10.91526055091492777321411932290