| L(s) = 1 | − 1.48e3·2-s + 5.90e4·3-s + 1.16e5·4-s − 8.78e7·6-s + 1.26e9·7-s + 2.94e9·8-s + 3.48e9·9-s + 4.25e10·11-s + 6.90e9·12-s − 7.14e11·13-s − 1.88e12·14-s − 4.62e12·16-s − 9.08e12·17-s − 5.18e12·18-s − 2.62e13·19-s + 7.48e13·21-s − 6.33e13·22-s + 2.96e13·23-s + 1.73e14·24-s + 1.06e15·26-s + 2.05e14·27-s + 1.48e14·28-s + 4.45e15·29-s + 4.93e14·31-s + 7.09e14·32-s + 2.51e15·33-s + 1.35e16·34-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.577·3-s + 0.0557·4-s − 0.593·6-s + 1.69·7-s + 0.970·8-s + 0.333·9-s + 0.495·11-s + 0.0321·12-s − 1.43·13-s − 1.74·14-s − 1.05·16-s − 1.09·17-s − 0.342·18-s − 0.983·19-s + 0.979·21-s − 0.508·22-s + 0.149·23-s + 0.560·24-s + 1.47·26-s + 0.192·27-s + 0.0946·28-s + 1.96·29-s + 0.108·31-s + 0.111·32-s + 0.285·33-s + 1.12·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.694291503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.694291503\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.48e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 1.26e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 4.25e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.14e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 9.08e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.62e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.96e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 4.45e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 4.93e14T + 2.08e31T^{2} \) |
| 37 | \( 1 + 8.82e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.17e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 4.77e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.81e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.01e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 8.26e17T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.60e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.57e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.32e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.66e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.85e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.68e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 6.58e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.13e21T + 5.27e41T^{2} \) |
| 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
−10.45432310929594688150197365427, −9.359482108014745199367067846945, −8.480542238854139106157420030096, −7.86752798393261177268267086695, −6.80735713819329200509574343002, −4.79878787603475283636371681302, −4.39238177739969696388592933140, −2.42694790482257175527798081887, −1.69130747885270809116816533325, −0.63422220520557733292249204716,
0.63422220520557733292249204716, 1.69130747885270809116816533325, 2.42694790482257175527798081887, 4.39238177739969696388592933140, 4.79878787603475283636371681302, 6.80735713819329200509574343002, 7.86752798393261177268267086695, 8.480542238854139106157420030096, 9.359482108014745199367067846945, 10.45432310929594688150197365427
