| L(s) = 1 | − 64·2-s − 156.·3-s + 4.09e3·4-s − 2.38e4·5-s + 1.00e4·6-s + 1.17e5·7-s − 2.62e5·8-s − 1.56e6·9-s + 1.52e6·10-s − 7.69e6·11-s − 6.42e5·12-s + 3.39e7·13-s − 7.52e6·14-s + 3.74e6·15-s + 1.67e7·16-s + 2.76e7·17-s + 1.00e8·18-s + 3.67e8·19-s − 9.77e7·20-s − 1.84e7·21-s + 4.92e8·22-s + 1.11e9·23-s + 4.11e7·24-s − 6.51e8·25-s − 2.17e9·26-s + 4.96e8·27-s + 4.81e8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.124·3-s + 0.5·4-s − 0.683·5-s + 0.0878·6-s + 0.377·7-s − 0.353·8-s − 0.984·9-s + 0.483·10-s − 1.30·11-s − 0.0621·12-s + 1.95·13-s − 0.267·14-s + 0.0848·15-s + 0.250·16-s + 0.278·17-s + 0.696·18-s + 1.79·19-s − 0.341·20-s − 0.0469·21-s + 0.925·22-s + 1.57·23-s + 0.0439·24-s − 0.533·25-s − 1.38·26-s + 0.246·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.040341072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040341072\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 1.17e5T \) |
| good | 3 | \( 1 + 156.T + 1.59e6T^{2} \) |
| 5 | \( 1 + 2.38e4T + 1.22e9T^{2} \) |
| 11 | \( 1 + 7.69e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.39e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 2.76e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.67e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.11e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 1.26e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 1.12e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 6.44e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.85e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.62e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 6.27e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.48e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.81e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.52e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.39e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 8.06e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.13e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.95e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.41e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 3.02e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.73e12T + 6.73e25T^{2} \) |
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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
−16.32967211634106424258565346577, −15.39665392869149713046844934397, −13.59597648215706240141784981397, −11.66861990635487541037459869418, −10.75944713425200007287647172559, −8.760093443543271052929621111782, −7.64182208453820525996811146247, −5.57890068998587350135203298024, −3.14378299594429020025744095199, −0.854299116562701416991740672694,
0.854299116562701416991740672694, 3.14378299594429020025744095199, 5.57890068998587350135203298024, 7.64182208453820525996811146247, 8.760093443543271052929621111782, 10.75944713425200007287647172559, 11.66861990635487541037459869418, 13.59597648215706240141784981397, 15.39665392869149713046844934397, 16.32967211634106424258565346577
