| L(s) = 1 | + 2.62e5·2-s + 6.56e8·3-s + 6.87e10·4-s + 4.54e12·5-s + 1.72e14·6-s + 3.07e15·7-s + 1.80e16·8-s − 1.86e16·9-s + 1.19e18·10-s + 1.92e19·11-s + 4.51e19·12-s − 7.06e20·13-s + 8.05e20·14-s + 2.98e21·15-s + 4.72e21·16-s + 6.97e22·17-s − 4.88e21·18-s + 6.63e23·19-s + 3.12e23·20-s + 2.01e24·21-s + 5.05e24·22-s + 1.23e25·23-s + 1.18e25·24-s − 5.21e25·25-s − 1.85e26·26-s − 3.08e26·27-s + 2.11e26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.979·3-s + 0.5·4-s + 0.532·5-s + 0.692·6-s + 0.712·7-s + 0.353·8-s − 0.0413·9-s + 0.376·10-s + 1.04·11-s + 0.489·12-s − 1.74·13-s + 0.504·14-s + 0.521·15-s + 0.250·16-s + 1.20·17-s − 0.0292·18-s + 1.46·19-s + 0.266·20-s + 0.698·21-s + 0.739·22-s + 0.796·23-s + 0.346·24-s − 0.716·25-s − 1.23·26-s − 1.01·27-s + 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(\approx\) |
\(4.463111029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.463111029\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.62e5T \) |
| good | 3 | \( 1 - 6.56e8T + 4.50e17T^{2} \) |
| 5 | \( 1 - 4.54e12T + 7.27e25T^{2} \) |
| 7 | \( 1 - 3.07e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 1.92e19T + 3.40e38T^{2} \) |
| 13 | \( 1 + 7.06e20T + 1.64e41T^{2} \) |
| 17 | \( 1 - 6.97e22T + 3.36e45T^{2} \) |
| 19 | \( 1 - 6.63e23T + 2.06e47T^{2} \) |
| 23 | \( 1 - 1.23e25T + 2.42e50T^{2} \) |
| 29 | \( 1 + 1.67e26T + 1.28e54T^{2} \) |
| 31 | \( 1 + 2.94e27T + 1.51e55T^{2} \) |
| 37 | \( 1 + 1.22e29T + 1.05e58T^{2} \) |
| 41 | \( 1 + 9.49e29T + 4.70e59T^{2} \) |
| 43 | \( 1 - 1.81e30T + 2.74e60T^{2} \) |
| 47 | \( 1 + 2.72e30T + 7.37e61T^{2} \) |
| 53 | \( 1 + 8.52e31T + 6.28e63T^{2} \) |
| 59 | \( 1 - 5.58e32T + 3.32e65T^{2} \) |
| 61 | \( 1 + 1.26e33T + 1.14e66T^{2} \) |
| 67 | \( 1 + 8.19e33T + 3.67e67T^{2} \) |
| 71 | \( 1 - 7.00e33T + 3.13e68T^{2} \) |
| 73 | \( 1 - 2.24e34T + 8.76e68T^{2} \) |
| 79 | \( 1 + 7.02e34T + 1.63e70T^{2} \) |
| 83 | \( 1 - 6.05e35T + 1.01e71T^{2} \) |
| 89 | \( 1 + 1.02e36T + 1.34e72T^{2} \) |
| 97 | \( 1 - 1.07e36T + 3.24e73T^{2} \) |
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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
−19.52918676248945698910236127338, −17.14710500777290908657244849836, −14.72920695550650551831062717891, −14.01021100532241977055645552609, −11.93759671991908892679954913749, −9.519283170510364101715421510297, −7.51659767810903556344756728415, −5.23412276945900515983514671119, −3.24531358055698392923568953908, −1.73529952177444601908514277158,
1.73529952177444601908514277158, 3.24531358055698392923568953908, 5.23412276945900515983514671119, 7.51659767810903556344756728415, 9.519283170510364101715421510297, 11.93759671991908892679954913749, 14.01021100532241977055645552609, 14.72920695550650551831062717891, 17.14710500777290908657244849836, 19.52918676248945698910236127338
