| L(s) = 1 | + 4.06·2-s + 3·3-s + 8.49·4-s + 16.2·5-s + 12.1·6-s + 6.77·7-s + 2.00·8-s + 9·9-s + 66.0·10-s − 16.5·11-s + 25.4·12-s − 82.3·13-s + 27.4·14-s + 48.8·15-s − 59.7·16-s + 6.48·17-s + 36.5·18-s + 151.·19-s + 138.·20-s + 20.3·21-s − 67.3·22-s − 154.·23-s + 6.02·24-s + 139.·25-s − 334.·26-s + 27·27-s + 57.5·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 0.577·3-s + 1.06·4-s + 1.45·5-s + 0.829·6-s + 0.365·7-s + 0.0888·8-s + 0.333·9-s + 2.08·10-s − 0.454·11-s + 0.613·12-s − 1.75·13-s + 0.524·14-s + 0.840·15-s − 0.934·16-s + 0.0924·17-s + 0.478·18-s + 1.82·19-s + 1.54·20-s + 0.211·21-s − 0.652·22-s − 1.39·23-s + 0.0512·24-s + 1.11·25-s − 2.52·26-s + 0.192·27-s + 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.915283867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.915283867\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 4.06T + 8T^{2} \) |
| 5 | \( 1 - 16.2T + 125T^{2} \) |
| 7 | \( 1 - 6.77T + 343T^{2} \) |
| 11 | \( 1 + 16.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.48T + 4.91e3T^{2} \) |
| 19 | \( 1 - 151.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 414.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 521.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 265.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 31.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 256.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 68.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 725.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 571.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 304.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 731.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 694.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 597.T + 9.12e5T^{2} \) |
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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
−12.51310727582915812542654095443, −11.69239756715870085483456773716, −10.01273482477152537854470711858, −9.567770104936104827031500839129, −7.937323650520057998937883583120, −6.65399104232796588515325085605, −5.44117438654571863888058235555, −4.78175984910134312931435393127, −3.05264810605709453813499907702, −2.07608455963124982518416107762,
2.07608455963124982518416107762, 3.05264810605709453813499907702, 4.78175984910134312931435393127, 5.44117438654571863888058235555, 6.65399104232796588515325085605, 7.937323650520057998937883583120, 9.567770104936104827031500839129, 10.01273482477152537854470711858, 11.69239756715870085483456773716, 12.51310727582915812542654095443
