| L(s) = 1 | − 64·2-s − 1.77e3·3-s + 4.09e3·4-s + 2.61e4·5-s + 1.13e5·6-s − 2.62e5·8-s + 1.56e6·9-s − 1.67e6·10-s + 1.12e7·11-s − 7.27e6·12-s − 1.74e7·13-s − 4.64e7·15-s + 1.67e7·16-s − 1.69e8·17-s − 9.99e7·18-s + 1.77e8·19-s + 1.07e8·20-s − 7.17e8·22-s − 4.47e7·23-s + 4.65e8·24-s − 5.37e8·25-s + 1.11e9·26-s + 5.85e7·27-s − 3.14e9·29-s + 2.97e9·30-s − 2.97e9·31-s − 1.07e9·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.40·3-s + 0.5·4-s + 0.748·5-s + 0.994·6-s − 0.353·8-s + 0.979·9-s − 0.529·10-s + 1.90·11-s − 0.703·12-s − 1.00·13-s − 1.05·15-s + 0.250·16-s − 1.70·17-s − 0.692·18-s + 0.867·19-s + 0.374·20-s − 1.34·22-s − 0.0630·23-s + 0.497·24-s − 0.440·25-s + 0.707·26-s + 0.0290·27-s − 0.981·29-s + 0.744·30-s − 0.601·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 64T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.77e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 2.61e4T + 1.22e9T^{2} \) |
| 11 | \( 1 - 1.12e7T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.74e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.69e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.77e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 4.47e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.14e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.97e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 5.72e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.70e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.32e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 3.33e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.03e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 2.88e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 2.11e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 5.92e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.43e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 6.46e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.80e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.71e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.63e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 7.42e12T + 6.73e25T^{2} \) |
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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
−10.96404602735096622769568863176, −9.684244290825609735293419411819, −9.087310292368606916561931527859, −7.21141860513078884487533537626, −6.41649066319732381402004782308, −5.53586296741899445828675160012, −4.18284821418994454134364619691, −2.17711061637322121961553374303, −1.09736235600723122721039103466, 0,
1.09736235600723122721039103466, 2.17711061637322121961553374303, 4.18284821418994454134364619691, 5.53586296741899445828675160012, 6.41649066319732381402004782308, 7.21141860513078884487533537626, 9.087310292368606916561931527859, 9.684244290825609735293419411819, 10.96404602735096622769568863176
