L(s) = 1 | + (2 + 3.46i)2-s + (−7.99 + 13.8i)4-s + (−37.7 − 65.3i)5-s + (99.4 + 83.1i)7-s − 63.9·8-s + (150. − 261. i)10-s + (−74.7 + 129. i)11-s + 349.·13-s + (−88.9 + 510. i)14-s + (−128 − 221. i)16-s + (−574. + 995. i)17-s + (1.39e3 + 2.42e3i)19-s + 1.20e3·20-s − 597.·22-s + (906. + 1.57e3i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.675 − 1.16i)5-s + (0.767 + 0.641i)7-s − 0.353·8-s + (0.477 − 0.826i)10-s + (−0.186 + 0.322i)11-s + 0.573·13-s + (−0.121 + 0.696i)14-s + (−0.125 − 0.216i)16-s + (−0.482 + 0.835i)17-s + (0.888 + 1.53i)19-s + 0.675·20-s − 0.263·22-s + (0.357 + 0.619i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.774437489\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.774437489\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 - 3.46i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-99.4 - 83.1i)T \) |
good | 5 | \( 1 + (37.7 + 65.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (74.7 - 129. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 349.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (574. - 995. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.39e3 - 2.42e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-906. - 1.57e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 759.T + 2.05e7T^{2} \) |
| 31 | \( 1 + (4.51e3 - 7.82e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.89e3 + 6.75e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 7.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.22e4 - 2.13e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-6.79e3 + 1.17e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.31e4 - 2.28e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.76e4 + 3.05e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.71e4 - 4.70e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 7.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.22e4 + 3.85e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.08e4 + 5.33e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 8.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-4.92e4 - 8.53e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 3.23e4T + 8.58e9T^{2} \) |
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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
−12.53446872502082297497047132326, −12.16731854174008987027063596343, −10.86875413472673608775950323597, −9.146089291219129019229596722986, −8.370005846194992190988347012092, −7.54417407631467050454184774319, −5.81834115349266830926858240154, −4.91503309142647855348890016267, −3.74275675750012345361824928925, −1.46635481624463237676080239482,
0.61348372369356748237067130245, 2.58068510479999918438822279089, 3.76025057644362189423688096054, 5.02111176747107316413682331182, 6.73206824287478074101875780849, 7.65066742977170227177663064754, 9.074350102716754245921414724597, 10.54997256087262906756088590123, 11.15636219001585365005270904404, 11.75177074711858353578277098089