L(s) = 1 | − 10.2i·2-s + (−4.60 + 14.8i)3-s − 72.9·4-s − 73.2·5-s + (152. + 47.1i)6-s + (55.7 − 117. i)7-s + 419. i·8-s + (−200. − 137. i)9-s + 751. i·10-s − 237. i·11-s + (335. − 1.08e3i)12-s − 63.3i·13-s + (−1.19e3 − 570. i)14-s + (337. − 1.09e3i)15-s + 1.96e3·16-s − 814.·17-s + ⋯ |
L(s) = 1 | − 1.81i·2-s + (−0.295 + 0.955i)3-s − 2.28·4-s − 1.31·5-s + (1.73 + 0.534i)6-s + (0.429 − 0.902i)7-s + 2.31i·8-s + (−0.825 − 0.563i)9-s + 2.37i·10-s − 0.593i·11-s + (0.672 − 2.17i)12-s − 0.104i·13-s + (−1.63 − 0.778i)14-s + (0.386 − 1.25i)15-s + 1.92·16-s − 0.683·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.152483 + 0.390889i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.152483 + 0.390889i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.60 - 14.8i)T \) |
| 7 | \( 1 + (-55.7 + 117. i)T \) |
good | 2 | \( 1 + 10.2iT - 32T^{2} \) |
| 5 | \( 1 + 73.2T + 3.12e3T^{2} \) |
| 11 | \( 1 + 237. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 63.3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 814.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 64.1iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 988. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.56e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.25e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.79e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.99e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.20e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.40e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 7.37e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.12e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.80e4iT - 8.58e9T^{2} \) |
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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
−16.46504554182653009012616001977, −14.89112267839370288778102595008, −13.34239177321392397896799045844, −11.58100329890728388092072322248, −11.21893210941193071543090582937, −9.945494723784323977674173631636, −8.331611061619992494353204727346, −4.53618401521729189447121888764, −3.53722294388708886316723898460, −0.32346105444179861885746270660,
4.88855566014675110696910846684, 6.57480302820617978215271775035, 7.75897159697716687383492350468, 8.678995645812360760464167281540, 11.65516506583917971550589833774, 12.91415902336027160621675677624, 14.51407389452783859441173751898, 15.40440706992517930367702634491, 16.47439157538131311104346177577, 17.80867869491232976289299855080