| L(s) = 1 | + 630.·2-s + 6.56e3·3-s + 2.66e5·4-s + 4.13e6·6-s + 2.83e6·7-s + 8.50e7·8-s + 4.30e7·9-s − 1.10e8·11-s + 1.74e9·12-s + 3.59e9·13-s + 1.78e9·14-s + 1.87e10·16-s + 3.08e10·17-s + 2.71e10·18-s − 7.55e10·19-s + 1.86e10·21-s − 6.93e10·22-s + 4.68e11·23-s + 5.58e11·24-s + 2.26e12·26-s + 2.82e11·27-s + 7.55e11·28-s − 4.60e11·29-s − 1.07e12·31-s + 6.63e11·32-s − 7.22e11·33-s + 1.94e13·34-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.577·3-s + 2.03·4-s + 1.00·6-s + 0.186·7-s + 1.79·8-s + 0.333·9-s − 0.154·11-s + 1.17·12-s + 1.22·13-s + 0.324·14-s + 1.09·16-s + 1.07·17-s + 0.580·18-s − 1.02·19-s + 0.107·21-s − 0.269·22-s + 1.24·23-s + 1.03·24-s + 2.12·26-s + 0.192·27-s + 0.377·28-s − 0.171·29-s − 0.226·31-s + 0.106·32-s − 0.0893·33-s + 1.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(10.13319329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.13319329\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 630.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 2.83e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.10e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.59e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.08e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 7.55e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 4.68e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 4.60e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 1.07e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.38e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 7.28e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.83e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 3.11e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.62e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.62e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.90e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 2.21e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 6.87e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 8.52e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.57e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.13e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.67e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 4.83e16T + 5.95e33T^{2} \) |
| 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
−11.47193405558071464817646085993, −10.54992907220516114631837259523, −8.891977475118208212167809972192, −7.60819565418376141564800966266, −6.41478558913451146847799751222, −5.42579227893232249674034952847, −4.25171082831248247568231506211, −3.40856380427934292614566765637, −2.42074289769904195467502384444, −1.17791575990309056079248682065,
1.17791575990309056079248682065, 2.42074289769904195467502384444, 3.40856380427934292614566765637, 4.25171082831248247568231506211, 5.42579227893232249674034952847, 6.41478558913451146847799751222, 7.60819565418376141564800966266, 8.891977475118208212167809972192, 10.54992907220516114631837259523, 11.47193405558071464817646085993
