| L(s) = 1 | + 6.55e4·2-s − 9.01e7·3-s + 4.29e9·4-s − 3.79e11·5-s − 5.90e12·6-s + 1.50e14·7-s + 2.81e14·8-s + 2.57e15·9-s − 2.48e16·10-s + 1.29e17·11-s − 3.87e17·12-s + 1.20e18·13-s + 9.89e18·14-s + 3.42e19·15-s + 1.84e19·16-s + 7.25e19·17-s + 1.68e20·18-s + 2.41e21·19-s − 1.63e21·20-s − 1.36e22·21-s + 8.50e21·22-s − 2.97e22·23-s − 2.53e22·24-s + 2.77e22·25-s + 7.92e22·26-s + 2.69e23·27-s + 6.48e23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.20·3-s + 0.5·4-s − 1.11·5-s − 0.855·6-s + 1.71·7-s + 0.353·8-s + 0.462·9-s − 0.786·10-s + 0.851·11-s − 0.604·12-s + 0.503·13-s + 1.21·14-s + 1.34·15-s + 0.250·16-s + 0.361·17-s + 0.327·18-s + 1.92·19-s − 0.556·20-s − 2.07·21-s + 0.602·22-s − 1.01·23-s − 0.427·24-s + 0.238·25-s + 0.356·26-s + 0.649·27-s + 0.858·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(1.937015703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.937015703\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 6.55e4T \) |
| good | 3 | \( 1 + 9.01e7T + 5.55e15T^{2} \) |
| 5 | \( 1 + 3.79e11T + 1.16e23T^{2} \) |
| 7 | \( 1 - 1.50e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 1.29e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 1.20e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 7.25e19T + 4.02e40T^{2} \) |
| 19 | \( 1 - 2.41e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.97e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.10e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 1.95e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 1.24e26T + 5.63e51T^{2} \) |
| 41 | \( 1 - 3.94e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 5.67e25T + 8.02e53T^{2} \) |
| 47 | \( 1 + 2.69e26T + 1.51e55T^{2} \) |
| 53 | \( 1 + 9.81e26T + 7.96e56T^{2} \) |
| 59 | \( 1 - 2.39e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 1.92e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 4.67e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 1.71e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 3.26e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 1.85e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 1.46e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.04e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.93e32T + 3.65e65T^{2} \) |
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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
−20.31590919979115517195144884960, −17.94302614026743016454597574653, −16.24586336183967891244723904353, −14.47359830577973256871647887775, −11.77033596894310440026522816141, −11.35670696479941157197490227224, −7.68497069964222198226327041299, −5.54247256695501639065762808126, −4.11358545594269031435804728242, −1.10233445261829048797961018763,
1.10233445261829048797961018763, 4.11358545594269031435804728242, 5.54247256695501639065762808126, 7.68497069964222198226327041299, 11.35670696479941157197490227224, 11.77033596894310440026522816141, 14.47359830577973256871647887775, 16.24586336183967891244723904353, 17.94302614026743016454597574653, 20.31590919979115517195144884960
