| L(s) = 1 | − 50.6·2-s + 243·3-s + 516.·4-s − 1.23e4·6-s − 2.44e4·7-s + 7.75e4·8-s + 5.90e4·9-s + 1.07e5·11-s + 1.25e5·12-s − 1.62e6·13-s + 1.23e6·14-s − 4.98e6·16-s + 3.92e6·17-s − 2.99e6·18-s + 4.71e6·19-s − 5.93e6·21-s − 5.42e6·22-s + 3.14e7·23-s + 1.88e7·24-s + 8.20e7·26-s + 1.43e7·27-s − 1.26e7·28-s − 2.17e8·29-s + 4.56e7·31-s + 9.37e7·32-s + 2.60e7·33-s − 1.98e8·34-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.577·3-s + 0.252·4-s − 0.646·6-s − 0.549·7-s + 0.836·8-s + 0.333·9-s + 0.200·11-s + 0.145·12-s − 1.21·13-s + 0.614·14-s − 1.18·16-s + 0.669·17-s − 0.373·18-s + 0.436·19-s − 0.317·21-s − 0.224·22-s + 1.01·23-s + 0.483·24-s + 1.35·26-s + 0.192·27-s − 0.138·28-s − 1.97·29-s + 0.286·31-s + 0.493·32-s + 0.115·33-s − 0.749·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.019813784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019813784\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 50.6T + 2.04e3T^{2} \) |
| 7 | \( 1 + 2.44e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.07e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.62e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.92e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.71e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.14e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.17e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 4.56e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.51e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.34e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.15e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.86e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.07e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 7.15e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 3.14e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 8.00e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.11e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.06e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.45e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.73e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.26e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.44e11T + 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.24074314038174757172997287823, −10.73266422385670762279667031646, −9.658461325269666006902843639445, −9.123110675547105181267426245395, −7.81251228406053661585652513394, −7.01385488339157386363585278065, −5.09704866281138466141948545103, −3.50652438821469899317124766433, −2.02164532617924264938315096094, −0.63094311769420123403213205196,
0.63094311769420123403213205196, 2.02164532617924264938315096094, 3.50652438821469899317124766433, 5.09704866281138466141948545103, 7.01385488339157386363585278065, 7.81251228406053661585652513394, 9.123110675547105181267426245395, 9.658461325269666006902843639445, 10.73266422385670762279667031646, 12.24074314038174757172997287823
