| L(s) = 1 | − 5.26·2-s + 25.5·3-s − 4.31·4-s − 134.·6-s + 131.·7-s + 191.·8-s + 408.·9-s + 290.·11-s − 110.·12-s + 68.3·13-s − 689.·14-s − 867.·16-s − 310.·17-s − 2.14e3·18-s − 2.13e3·19-s + 3.34e3·21-s − 1.52e3·22-s + 873.·23-s + 4.87e3·24-s − 359.·26-s + 4.22e3·27-s − 564.·28-s − 2.58e3·29-s − 9.08e3·31-s − 1.54e3·32-s + 7.40e3·33-s + 1.63e3·34-s + ⋯ |
| L(s) = 1 | − 0.930·2-s + 1.63·3-s − 0.134·4-s − 1.52·6-s + 1.01·7-s + 1.05·8-s + 1.68·9-s + 0.722·11-s − 0.220·12-s + 0.112·13-s − 0.940·14-s − 0.847·16-s − 0.260·17-s − 1.56·18-s − 1.35·19-s + 1.65·21-s − 0.672·22-s + 0.344·23-s + 1.72·24-s − 0.104·26-s + 1.11·27-s − 0.136·28-s − 0.569·29-s − 1.69·31-s − 0.267·32-s + 1.18·33-s + 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.508432894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508432894\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 5.26T + 32T^{2} \) |
| 3 | \( 1 - 25.5T + 243T^{2} \) |
| 7 | \( 1 - 131.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 68.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 310.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.13e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 873.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.69e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.23e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.32e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.18e4T + 8.58e9T^{2} \) |
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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
−16.76782391118179749278291581339, −14.99655330484950880911238714015, −14.25298072430603815614484980075, −13.06354839012604626163844270933, −10.87458170338699503408965297617, −9.296757616601543303901014697057, −8.572227182224539262010358264535, −7.49507376360073757576428114929, −4.18882828830997964646998933624, −1.79831899151529647478138098756,
1.79831899151529647478138098756, 4.18882828830997964646998933624, 7.49507376360073757576428114929, 8.572227182224539262010358264535, 9.296757616601543303901014697057, 10.87458170338699503408965297617, 13.06354839012604626163844270933, 14.25298072430603815614484980075, 14.99655330484950880911238714015, 16.76782391118179749278291581339
