| L(s) = 1 | + 2.23·2-s − 1.23·3-s + 3.00·4-s + 0.618·5-s − 2.76·6-s − 2.61·7-s + 2.23·8-s − 1.47·9-s + 1.38·10-s − 3.70·12-s + 6.47·13-s − 5.85·14-s − 0.763·15-s − 0.999·16-s + 8.09·17-s − 3.29·18-s + 19-s + 1.85·20-s + 3.23·21-s + 1.61·23-s − 2.76·24-s − 4.61·25-s + 14.4·26-s + 5.52·27-s − 7.85·28-s − 0.472·29-s − 1.70·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.713·3-s + 1.50·4-s + 0.276·5-s − 1.12·6-s − 0.989·7-s + 0.790·8-s − 0.490·9-s + 0.437·10-s − 1.07·12-s + 1.79·13-s − 1.56·14-s − 0.197·15-s − 0.249·16-s + 1.96·17-s − 0.775·18-s + 0.229·19-s + 0.414·20-s + 0.706·21-s + 0.337·23-s − 0.564·24-s − 0.923·25-s + 2.83·26-s + 1.06·27-s − 1.48·28-s − 0.0876·29-s − 0.311·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.414289893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.414289893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 - 8.09T + 17T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 - 8.09T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 3.52T + 59T^{2} \) |
| 61 | \( 1 - 6.38T + 61T^{2} \) |
| 67 | \( 1 - 0.472T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 1.23T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
| 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
−9.040363479089810858807368543944, −8.121702973811623717606672642092, −6.96602839200597526463022295762, −6.22000145760170726518582657175, −5.75474493766509758148303136094, −5.35112146065011846909959060742, −4.06320088520080266305743706101, −3.43684680778838777387181267273, −2.69020502529472296104486305512, −1.01561239181170106658328356964,
1.01561239181170106658328356964, 2.69020502529472296104486305512, 3.43684680778838777387181267273, 4.06320088520080266305743706101, 5.35112146065011846909959060742, 5.75474493766509758148303136094, 6.22000145760170726518582657175, 6.96602839200597526463022295762, 8.121702973811623717606672642092, 9.040363479089810858807368543944
