L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 90.8·5-s − 36·6-s + 19.2·7-s − 64·8-s + 81·9-s − 363.·10-s − 68.0·11-s + 144·12-s + 977.·13-s − 77.1·14-s + 817.·15-s + 256·16-s − 635.·17-s − 324·18-s − 2.26e3·19-s + 1.45e3·20-s + 173.·21-s + 272.·22-s + 4.98e3·23-s − 576·24-s + 5.11e3·25-s − 3.91e3·26-s + 729·27-s + 308.·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.62·5-s − 0.408·6-s + 0.148·7-s − 0.353·8-s + 0.333·9-s − 1.14·10-s − 0.169·11-s + 0.288·12-s + 1.60·13-s − 0.105·14-s + 0.937·15-s + 0.250·16-s − 0.533·17-s − 0.235·18-s − 1.44·19-s + 0.812·20-s + 0.0859·21-s + 0.119·22-s + 1.96·23-s − 0.204·24-s + 1.63·25-s − 1.13·26-s + 0.192·27-s + 0.0744·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.943424966\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.943424966\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
good | 5 | \( 1 - 90.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 19.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 68.0T + 1.61e5T^{2} \) |
| 13 | \( 1 - 977.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 635.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 792.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 85.1T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.64e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.44e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.73e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 5.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.86e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.11e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.66e5T + 8.58e9T^{2} \) |
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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
−10.62903702642736679523311521203, −9.513648892139149301167012496398, −8.893899785509439845479491767969, −8.185509175499620040722922217762, −6.65736390955060893869874763701, −6.16569670877803933462328291410, −4.77036413341075836800818184901, −3.08101222642242072085131275188, −2.01946004657108576300166964154, −1.09614368871244646810692162557,
1.09614368871244646810692162557, 2.01946004657108576300166964154, 3.08101222642242072085131275188, 4.77036413341075836800818184901, 6.16569670877803933462328291410, 6.65736390955060893869874763701, 8.185509175499620040722922217762, 8.893899785509439845479491767969, 9.513648892139149301167012496398, 10.62903702642736679523311521203