L(s) = 1 | + (3.99 − 6.92i)3-s + (31.6 − 54.7i)5-s + 80.2·7-s + (89.5 + 155. i)9-s + 475.·11-s + (−337. − 584. i)13-s + (−252. − 437. i)15-s + (866. − 1.50e3i)17-s + (574. + 1.46e3i)19-s + (320. − 555. i)21-s + (2.42e3 + 4.19e3i)23-s + (−435. − 754. i)25-s + 3.37e3·27-s + (2.39e3 + 4.14e3i)29-s + 127.·31-s + ⋯ |
L(s) = 1 | + (0.256 − 0.444i)3-s + (0.565 − 0.979i)5-s + 0.619·7-s + (0.368 + 0.638i)9-s + 1.18·11-s + (−0.553 − 0.959i)13-s + (−0.289 − 0.502i)15-s + (0.727 − 1.25i)17-s + (0.365 + 0.931i)19-s + (0.158 − 0.274i)21-s + (0.955 + 1.65i)23-s + (−0.139 − 0.241i)25-s + 0.890·27-s + (0.528 + 0.915i)29-s + 0.0237·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.247461805\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.247461805\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-574. - 1.46e3i)T \) |
good | 3 | \( 1 + (-3.99 + 6.92i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-31.6 + 54.7i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 80.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 475.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (337. + 584. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-866. + 1.50e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-2.42e3 - 4.19e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.39e3 - 4.14e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 127.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.88e3 + 1.36e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (964. - 1.66e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (8.09e3 + 1.40e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-8.92e3 - 1.54e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (8.42e3 - 1.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.16e4 - 2.01e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.36e4 + 2.35e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.74e4 + 6.48e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-3.49e4 + 6.04e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.14e4 - 1.97e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 5.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-8.20e3 - 1.42e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-6.57e4 + 1.13e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−10.67515839106875594215194411731, −9.620280602875824622981207744495, −8.884835767010764947844749468313, −7.79564866047333923913790973506, −7.09412859203820168453173660435, −5.41651743239430578135878554900, −4.98902332191193000856572337447, −3.33145619888897442337663606992, −1.70914007642731485798345192246, −1.04341915670924118842273705649,
1.23047015058913751721594322106, 2.56256614339299967012516869530, 3.81643955700153674977989579146, 4.80059861318094465233774957875, 6.49625294131101923390314826082, 6.76726159415583397564823981984, 8.327225193976851480361609240023, 9.305947095110584317093279135667, 10.02119047516016626193858751858, 10.93050977988238976582716463618