L(s) = 1 | − 6.05·2-s − 15.9·3-s + 4.62·4-s − 25·5-s + 96.6·6-s + 165.·8-s + 12.2·9-s + 151.·10-s + 708.·11-s − 73.8·12-s − 1.12e3·13-s + 399.·15-s − 1.15e3·16-s + 393.·17-s − 73.9·18-s − 1.79e3·19-s − 115.·20-s − 4.28e3·22-s − 3.57e3·23-s − 2.64e3·24-s + 625·25-s + 6.80e3·26-s + 3.68e3·27-s − 2.14·29-s − 2.41e3·30-s − 8.77e3·31-s + 1.66e3·32-s + ⋯ |
L(s) = 1 | − 1.06·2-s − 1.02·3-s + 0.144·4-s − 0.447·5-s + 1.09·6-s + 0.915·8-s + 0.0502·9-s + 0.478·10-s + 1.76·11-s − 0.148·12-s − 1.84·13-s + 0.458·15-s − 1.12·16-s + 0.330·17-s − 0.0537·18-s − 1.13·19-s − 0.0646·20-s − 1.88·22-s − 1.41·23-s − 0.937·24-s + 0.200·25-s + 1.97·26-s + 0.973·27-s − 0.000474·29-s − 0.490·30-s − 1.63·31-s + 0.286·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2290440368\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2290440368\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 6.05T + 32T^{2} \) |
| 3 | \( 1 + 15.9T + 243T^{2} \) |
| 11 | \( 1 - 708.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 393.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.14T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.81e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.20e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.87e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.25e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.12e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.54e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.97e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.97e4T + 8.58e9T^{2} \) |
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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
−11.18516614800471445451045682832, −10.16554462918299175729688786282, −9.391996738530907213453279125586, −8.405497174411177911658556553578, −7.30686630170569941105557147677, −6.41761379473763646029645578164, −5.03214460040245531016614314070, −3.99329301763273863964942439298, −1.81881897729783519785762245566, −0.34625618278816309167364647218,
0.34625618278816309167364647218, 1.81881897729783519785762245566, 3.99329301763273863964942439298, 5.03214460040245531016614314070, 6.41761379473763646029645578164, 7.30686630170569941105557147677, 8.405497174411177911658556553578, 9.391996738530907213453279125586, 10.16554462918299175729688786282, 11.18516614800471445451045682832