| L(s) = 1 | + 4.53·2-s − 10.5·3-s − 11.4·4-s − 25·5-s − 47.9·6-s − 49·7-s − 196.·8-s − 130.·9-s − 113.·10-s + 90.5·11-s + 121.·12-s − 74.8·13-s − 222.·14-s + 264.·15-s − 525.·16-s + 1.03e3·17-s − 592.·18-s − 31.6·19-s + 286.·20-s + 519.·21-s + 410.·22-s − 3.85e3·23-s + 2.08e3·24-s + 625·25-s − 339.·26-s + 3.95e3·27-s + 561.·28-s + ⋯ |
| L(s) = 1 | + 0.800·2-s − 0.679·3-s − 0.358·4-s − 0.447·5-s − 0.544·6-s − 0.377·7-s − 1.08·8-s − 0.538·9-s − 0.358·10-s + 0.225·11-s + 0.243·12-s − 0.122·13-s − 0.302·14-s + 0.303·15-s − 0.513·16-s + 0.866·17-s − 0.431·18-s − 0.0201·19-s + 0.160·20-s + 0.256·21-s + 0.180·22-s − 1.52·23-s + 0.739·24-s + 0.200·25-s − 0.0983·26-s + 1.04·27-s + 0.135·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 2 | \( 1 - 4.53T + 32T^{2} \) |
| 3 | \( 1 + 10.5T + 243T^{2} \) |
| 11 | \( 1 - 90.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 74.8T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 31.6T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 866.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.92e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.18e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.03e3T + 8.58e9T^{2} \) |
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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
−14.73441856505347053689854138156, −13.74170258001829973367498041817, −12.35807842413409799800317033984, −11.67654231930691529753016475398, −9.975824195018469546036007313656, −8.365465018255132161276631766095, −6.34587109037505171982306508693, −5.10370852580996760227812753283, −3.48907564507549800331574950178, 0,
3.48907564507549800331574950178, 5.10370852580996760227812753283, 6.34587109037505171982306508693, 8.365465018255132161276631766095, 9.975824195018469546036007313656, 11.67654231930691529753016475398, 12.35807842413409799800317033984, 13.74170258001829973367498041817, 14.73441856505347053689854138156
