| L(s) = 1 | − 10.6·2-s − 9·3-s + 80.6·4-s − 25·5-s + 95.5·6-s + 32.7·7-s − 515.·8-s + 81·9-s + 265.·10-s − 121·11-s − 725.·12-s − 345.·13-s − 347.·14-s + 225·15-s + 2.89e3·16-s − 1.78e3·17-s − 859.·18-s + 2.40e3·19-s − 2.01e3·20-s − 294.·21-s + 1.28e3·22-s + 4.87e3·23-s + 4.64e3·24-s + 625·25-s + 3.66e3·26-s − 729·27-s + 2.63e3·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.51·4-s − 0.447·5-s + 1.08·6-s + 0.252·7-s − 2.84·8-s + 0.333·9-s + 0.838·10-s − 0.301·11-s − 1.45·12-s − 0.567·13-s − 0.473·14-s + 0.258·15-s + 2.82·16-s − 1.50·17-s − 0.625·18-s + 1.52·19-s − 1.12·20-s − 0.145·21-s + 0.565·22-s + 1.92·23-s + 1.64·24-s + 0.200·25-s + 1.06·26-s − 0.192·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 10.6T + 32T^{2} \) |
| 7 | \( 1 - 32.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 345.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.40e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 458.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.08e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.65e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.51e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.17e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−11.28065512613894117082123011597, −10.33862592071702959736057238842, −9.369894278511904821120178489908, −8.426014216946454056832712799940, −7.35206124825065248069851600646, −6.66783072387207929125018653415, −5.01791634271400013331133080510, −2.79089897528618167814636754196, −1.20163498071949478550271471707, 0,
1.20163498071949478550271471707, 2.79089897528618167814636754196, 5.01791634271400013331133080510, 6.66783072387207929125018653415, 7.35206124825065248069851600646, 8.426014216946454056832712799940, 9.369894278511904821120178489908, 10.33862592071702959736057238842, 11.28065512613894117082123011597
