| L(s) = 1 | + 31.0·2-s − 288.·3-s − 1.08e3·4-s − 6.16e3·5-s − 8.94e3·6-s + 1.68e4·7-s − 9.72e4·8-s − 9.40e4·9-s − 1.91e5·10-s − 1.33e5·11-s + 3.12e5·12-s + 1.72e6·13-s + 5.21e5·14-s + 1.77e6·15-s − 7.97e5·16-s + 1.90e6·17-s − 2.91e6·18-s − 1.35e7·19-s + 6.68e6·20-s − 4.84e6·21-s − 4.15e6·22-s − 3.53e7·23-s + 2.80e7·24-s − 1.07e7·25-s + 5.36e7·26-s + 7.81e7·27-s − 1.82e7·28-s + ⋯ |
| L(s) = 1 | + 0.685·2-s − 0.684·3-s − 0.529·4-s − 0.882·5-s − 0.469·6-s + 0.377·7-s − 1.04·8-s − 0.530·9-s − 0.605·10-s − 0.250·11-s + 0.362·12-s + 1.29·13-s + 0.259·14-s + 0.604·15-s − 0.190·16-s + 0.324·17-s − 0.364·18-s − 1.25·19-s + 0.467·20-s − 0.258·21-s − 0.172·22-s − 1.14·23-s + 0.718·24-s − 0.220·25-s + 0.885·26-s + 1.04·27-s − 0.200·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 1.68e4T \) |
| good | 2 | \( 1 - 31.0T + 2.04e3T^{2} \) |
| 3 | \( 1 + 288.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 6.16e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 1.33e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.72e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.35e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.53e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 8.11e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 8.72e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.94e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.60e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.44e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.49e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.86e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.50e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.08e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.67e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 8.61e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.88e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.42e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.68e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.34e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.43e10T + 7.15e21T^{2} \) |
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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
−18.81276157193396313880207906295, −17.44837445569119936302030383152, −15.70646079649871382256335496444, −14.14395809131426295990184866016, −12.43924517480024763449603333298, −11.09305020580375350138968234729, −8.440986099086814035453753842598, −5.77480118422244121392872850335, −3.96073224635113559685749478845, 0,
3.96073224635113559685749478845, 5.77480118422244121392872850335, 8.440986099086814035453753842598, 11.09305020580375350138968234729, 12.43924517480024763449603333298, 14.14395809131426295990184866016, 15.70646079649871382256335496444, 17.44837445569119936302030383152, 18.81276157193396313880207906295
