| L(s) = 1 | − 1.89·2-s − 2.95·3-s − 4.42·4-s + 5.58·6-s − 5.94·7-s + 23.4·8-s − 18.2·9-s + 11.5·11-s + 13.0·12-s − 22.6·13-s + 11.2·14-s − 9.02·16-s − 120.·17-s + 34.5·18-s − 19·19-s + 17.5·21-s − 21.7·22-s − 63.9·23-s − 69.3·24-s + 42.7·26-s + 133.·27-s + 26.2·28-s − 89.7·29-s − 251.·31-s − 170.·32-s − 34.0·33-s + 227.·34-s + ⋯ |
| L(s) = 1 | − 0.668·2-s − 0.568·3-s − 0.553·4-s + 0.380·6-s − 0.320·7-s + 1.03·8-s − 0.676·9-s + 0.315·11-s + 0.314·12-s − 0.482·13-s + 0.214·14-s − 0.140·16-s − 1.71·17-s + 0.452·18-s − 0.229·19-s + 0.182·21-s − 0.211·22-s − 0.579·23-s − 0.590·24-s + 0.322·26-s + 0.953·27-s + 0.177·28-s − 0.574·29-s − 1.45·31-s − 0.944·32-s − 0.179·33-s + 1.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3758122930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3758122930\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 1.89T + 8T^{2} \) |
| 3 | \( 1 + 2.95T + 27T^{2} \) |
| 7 | \( 1 + 5.94T + 343T^{2} \) |
| 11 | \( 1 - 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 63.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 373.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 364.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 376.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 816.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 220.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 383.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 537.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 616.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 524.T + 9.12e5T^{2} \) |
| 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
−10.74517060590801912081719777041, −9.436482581375267902912533771777, −9.043350233772598800900331803727, −8.017505772753163634616575100963, −6.97102109035760990027347337903, −5.93973126883544044364403939594, −4.89325207441706797424285493401, −3.86676459926048703157287418771, −2.15506696931849794623376472593, −0.42241668340230481051776815580,
0.42241668340230481051776815580, 2.15506696931849794623376472593, 3.86676459926048703157287418771, 4.89325207441706797424285493401, 5.93973126883544044364403939594, 6.97102109035760990027347337903, 8.017505772753163634616575100963, 9.043350233772598800900331803727, 9.436482581375267902912533771777, 10.74517060590801912081719777041
