| L(s) = 1 | − 1.20·2-s − 0.200·3-s − 0.557·4-s − 3.21·5-s + 0.241·6-s + 7-s + 3.07·8-s − 2.95·9-s + 3.86·10-s + 4.57·11-s + 0.112·12-s + 0.703·13-s − 1.20·14-s + 0.646·15-s − 2.57·16-s + 4.25·17-s + 3.55·18-s + 8.04·19-s + 1.79·20-s − 0.200·21-s − 5.49·22-s − 5.34·23-s − 0.617·24-s + 5.34·25-s − 0.844·26-s + 1.19·27-s − 0.557·28-s + ⋯ |
| L(s) = 1 | − 0.849·2-s − 0.116·3-s − 0.278·4-s − 1.43·5-s + 0.0985·6-s + 0.377·7-s + 1.08·8-s − 0.986·9-s + 1.22·10-s + 1.38·11-s + 0.0323·12-s + 0.194·13-s − 0.320·14-s + 0.166·15-s − 0.643·16-s + 1.03·17-s + 0.837·18-s + 1.84·19-s + 0.401·20-s − 0.0438·21-s − 1.17·22-s − 1.11·23-s − 0.126·24-s + 1.06·25-s − 0.165·26-s + 0.230·27-s − 0.105·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5803439310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5803439310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.20T + 2T^{2} \) |
| 3 | \( 1 + 0.200T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - 0.703T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 8.04T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 31 | \( 1 - 7.61T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 + 3.09T + 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 - 4.13T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
| 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
−11.70219060275614592477279220848, −11.01826488670211672961083216078, −9.706629744644781310948286293569, −8.927569599322396301055577380190, −7.922897850881759007817721317396, −7.51780407807136346521202609439, −5.85861918418398756981625325491, −4.45156554836848153579807534973, −3.46635290343755547965433793262, −0.947552940638209380269399478920,
0.947552940638209380269399478920, 3.46635290343755547965433793262, 4.45156554836848153579807534973, 5.85861918418398756981625325491, 7.51780407807136346521202609439, 7.922897850881759007817721317396, 8.927569599322396301055577380190, 9.706629744644781310948286293569, 11.01826488670211672961083216078, 11.70219060275614592477279220848
