| L(s) = 1 | − 4.84·2-s + 9.58·3-s + 15.4·4-s − 46.3·6-s − 7·7-s − 36.0·8-s + 64.7·9-s + 62.1·11-s + 147.·12-s − 14.0·13-s + 33.8·14-s + 50.9·16-s − 63.5·17-s − 313.·18-s + 48.7·19-s − 67.0·21-s − 301.·22-s + 99.3·23-s − 345.·24-s + 68.2·26-s + 362.·27-s − 108.·28-s − 69.0·29-s − 9.68·31-s + 41.7·32-s + 595.·33-s + 307.·34-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.84·3-s + 1.93·4-s − 3.15·6-s − 0.377·7-s − 1.59·8-s + 2.39·9-s + 1.70·11-s + 3.55·12-s − 0.300·13-s + 0.646·14-s + 0.795·16-s − 0.906·17-s − 4.10·18-s + 0.588·19-s − 0.696·21-s − 2.91·22-s + 0.900·23-s − 2.93·24-s + 0.514·26-s + 2.58·27-s − 0.729·28-s − 0.442·29-s − 0.0561·31-s + 0.230·32-s + 3.14·33-s + 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.544092944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.544092944\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 4.84T + 8T^{2} \) |
| 3 | \( 1 - 9.58T + 27T^{2} \) |
| 11 | \( 1 - 62.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 63.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.68T + 2.97e4T^{2} \) |
| 37 | \( 1 + 240.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 51.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 451.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 268.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 323.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 541.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 161.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 305.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 504.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 513.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 543.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.86e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.09283125001758694192565456948, −10.80223475511244050995761026133, −9.592529723979436201608869094205, −9.183737465872716343535020810613, −8.556627141196053326647380862039, −7.37911752003919553288537713499, −6.75822527144168371605956012013, −3.94033848476198873690756859061, −2.57098019204571101580261907459, −1.31829429226928772840387860829,
1.31829429226928772840387860829, 2.57098019204571101580261907459, 3.94033848476198873690756859061, 6.75822527144168371605956012013, 7.37911752003919553288537713499, 8.556627141196053326647380862039, 9.183737465872716343535020810613, 9.592529723979436201608869094205, 10.80223475511244050995761026133, 12.09283125001758694192565456948
