| L(s) = 1 | + 4·2-s + 28.0·3-s + 16·4-s − 31.0·5-s + 112.·6-s + 14.7·7-s + 64·8-s + 542.·9-s − 124.·10-s + 0.310·11-s + 448.·12-s + 699.·13-s + 59.1·14-s − 869.·15-s + 256·16-s − 1.22e3·17-s + 2.17e3·18-s − 701.·19-s − 496.·20-s + 414.·21-s + 1.24·22-s − 2.75e3·23-s + 1.79e3·24-s − 2.16e3·25-s + 2.79e3·26-s + 8.40e3·27-s + 236.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.79·3-s + 0.5·4-s − 0.554·5-s + 1.27·6-s + 0.113·7-s + 0.353·8-s + 2.23·9-s − 0.392·10-s + 0.000773·11-s + 0.899·12-s + 1.14·13-s + 0.0805·14-s − 0.997·15-s + 0.250·16-s − 1.03·17-s + 1.57·18-s − 0.445·19-s − 0.277·20-s + 0.204·21-s + 0.000546·22-s − 1.08·23-s + 0.635·24-s − 0.692·25-s + 0.812·26-s + 2.21·27-s + 0.0569·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.397987368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.397987368\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 37 | \( 1 + 1.36e3T \) |
| good | 3 | \( 1 - 28.0T + 243T^{2} \) |
| 5 | \( 1 + 31.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 14.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 0.310T + 1.61e5T^{2} \) |
| 13 | \( 1 - 699.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 701.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.44e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + 1.91e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.72e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.79e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.57e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.45e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.31e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.14e4T + 8.58e9T^{2} \) |
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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
−13.68728983385600010496342542858, −12.95628639185684783689301871403, −11.56448902232000821402369885156, −10.13690665494851137406028849838, −8.676584012830275600583717105797, −7.978512512658011078879882459553, −6.58326386559907063228836810047, −4.34508203116806494244058070937, −3.40821172019087673500560295635, −1.95631710291740471464130684684,
1.95631710291740471464130684684, 3.40821172019087673500560295635, 4.34508203116806494244058070937, 6.58326386559907063228836810047, 7.978512512658011078879882459553, 8.676584012830275600583717105797, 10.13690665494851137406028849838, 11.56448902232000821402369885156, 12.95628639185684783689301871403, 13.68728983385600010496342542858
