L(s) = 1 | + (−0.841 − 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (0.797 + 0.234i)5-s + (−0.415 + 0.909i)6-s + (−0.654 + 0.755i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.544 − 0.627i)10-s + (1.10 − 0.708i)11-s + (0.841 − 0.540i)12-s + (0.959 − 0.281i)14-s + (0.118 − 0.822i)15-s + (−0.654 + 0.755i)16-s + (0.118 − 0.258i)17-s + (0.959 + 0.281i)18-s + ⋯ |
L(s) = 1 | + (−0.841 − 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (0.797 + 0.234i)5-s + (−0.415 + 0.909i)6-s + (−0.654 + 0.755i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.544 − 0.627i)10-s + (1.10 − 0.708i)11-s + (0.841 − 0.540i)12-s + (0.959 − 0.281i)14-s + (0.118 − 0.822i)15-s + (−0.654 + 0.755i)16-s + (0.118 − 0.258i)17-s + (0.959 + 0.281i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8315452671\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8315452671\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (0.415 + 0.909i)T \) |
good | 5 | \( 1 + (-0.797 - 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 11 | \( 1 + (-1.10 + 0.708i)T + (0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-1.61 - 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 61 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 89 | \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
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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
−9.462430643115743162177662830685, −8.456554260829189064559180494183, −7.893562703653225983555446600606, −6.88025739454192603429209951669, −6.05882933421651606337272654538, −5.86640758402313288788516120100, −3.95262113205066515837825680182, −2.88397305150120862235149302742, −2.17584526383726593858372646319, −1.06415292633846106951842680058,
1.12325817816109818516164024613, 2.58163261470369493850481160404, 3.88524320982884505809294824215, 4.77230547769784970157601655131, 5.68334125465438319799432258088, 6.38461068971317652626152061114, 7.11362774917218831591567997384, 8.023694935598836010965033192160, 9.295589176269763876486773008877, 9.418940751061957591447121998034