| L(s) = 1 | − 171.·2-s − 729·3-s + 2.11e4·4-s + 1.24e5·6-s + 1.62e5·7-s − 2.22e6·8-s + 5.31e5·9-s + 6.40e6·11-s − 1.54e7·12-s − 2.02e7·13-s − 2.77e7·14-s + 2.08e8·16-s + 1.34e8·17-s − 9.11e7·18-s + 1.95e8·19-s − 1.18e8·21-s − 1.09e9·22-s + 1.34e9·23-s + 1.62e9·24-s + 3.46e9·26-s − 3.87e8·27-s + 3.43e9·28-s + 4.25e9·29-s + 5.99e9·31-s − 1.74e10·32-s − 4.66e9·33-s − 2.30e10·34-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.577·3-s + 2.58·4-s + 1.09·6-s + 0.520·7-s − 3.00·8-s + 0.333·9-s + 1.09·11-s − 1.49·12-s − 1.16·13-s − 0.986·14-s + 3.10·16-s + 1.34·17-s − 0.631·18-s + 0.954·19-s − 0.300·21-s − 2.06·22-s + 1.89·23-s + 1.73·24-s + 2.20·26-s − 0.192·27-s + 1.34·28-s + 1.32·29-s + 1.21·31-s − 2.87·32-s − 0.629·33-s − 2.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.9322111704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9322111704\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 729T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 171.T + 8.19e3T^{2} \) |
| 7 | \( 1 - 1.62e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 6.40e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.02e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.34e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.95e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.34e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.25e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 5.99e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.63e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.26e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.75e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 9.16e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.75e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 3.03e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.25e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 2.29e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.77e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 6.66e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.73e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.27e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 4.87e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 4.88e12T + 6.73e25T^{2} \) |
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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
−11.71983123444254421309714350322, −10.46983793697610532060708087960, −9.700467607708680593799841834380, −8.649500959347223138151865359903, −7.45019853131763193223757799833, −6.69785691833331156788675291834, −5.17321737128865217680599933920, −2.96299652684598594888876457792, −1.41121534497906276126214097037, −0.76018872819097900813837134126,
0.76018872819097900813837134126, 1.41121534497906276126214097037, 2.96299652684598594888876457792, 5.17321737128865217680599933920, 6.69785691833331156788675291834, 7.45019853131763193223757799833, 8.649500959347223138151865359903, 9.700467607708680593799841834380, 10.46983793697610532060708087960, 11.71983123444254421309714350322
