| L(s) = 1 | + 4.28e5·2-s + 3.55e8·3-s + 4.57e10·4-s + 1.52e14·6-s − 7.08e15·7-s − 3.92e16·8-s − 3.23e17·9-s + 1.42e19·11-s + 1.62e19·12-s − 1.67e20·13-s − 3.03e21·14-s − 2.30e22·16-s − 1.07e23·17-s − 1.38e23·18-s + 7.72e23·19-s − 2.52e24·21-s + 6.08e24·22-s + 1.48e25·23-s − 1.39e25·24-s − 7.16e25·26-s − 2.75e26·27-s − 3.24e26·28-s − 1.24e26·29-s − 6.93e27·31-s − 4.48e27·32-s + 5.05e27·33-s − 4.58e28·34-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.530·3-s + 0.333·4-s + 0.612·6-s − 1.64·7-s − 0.770·8-s − 0.718·9-s + 0.770·11-s + 0.176·12-s − 0.412·13-s − 1.89·14-s − 1.22·16-s − 1.84·17-s − 0.830·18-s + 1.70·19-s − 0.872·21-s + 0.889·22-s + 0.955·23-s − 0.408·24-s − 0.476·26-s − 0.911·27-s − 0.547·28-s − 0.109·29-s − 1.78·31-s − 0.640·32-s + 0.408·33-s − 2.13·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(\approx\) |
\(2.101926318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101926318\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 4.28e5T + 1.37e11T^{2} \) |
| 3 | \( 1 - 3.55e8T + 4.50e17T^{2} \) |
| 7 | \( 1 + 7.08e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 1.42e19T + 3.40e38T^{2} \) |
| 13 | \( 1 + 1.67e20T + 1.64e41T^{2} \) |
| 17 | \( 1 + 1.07e23T + 3.36e45T^{2} \) |
| 19 | \( 1 - 7.72e23T + 2.06e47T^{2} \) |
| 23 | \( 1 - 1.48e25T + 2.42e50T^{2} \) |
| 29 | \( 1 + 1.24e26T + 1.28e54T^{2} \) |
| 31 | \( 1 + 6.93e27T + 1.51e55T^{2} \) |
| 37 | \( 1 - 3.14e28T + 1.05e58T^{2} \) |
| 41 | \( 1 + 5.42e29T + 4.70e59T^{2} \) |
| 43 | \( 1 - 6.71e29T + 2.74e60T^{2} \) |
| 47 | \( 1 - 1.17e31T + 7.37e61T^{2} \) |
| 53 | \( 1 + 6.78e30T + 6.28e63T^{2} \) |
| 59 | \( 1 + 7.22e32T + 3.32e65T^{2} \) |
| 61 | \( 1 + 2.71e32T + 1.14e66T^{2} \) |
| 67 | \( 1 - 1.11e34T + 3.67e67T^{2} \) |
| 71 | \( 1 - 4.45e33T + 3.13e68T^{2} \) |
| 73 | \( 1 - 1.80e34T + 8.76e68T^{2} \) |
| 79 | \( 1 + 2.04e34T + 1.63e70T^{2} \) |
| 83 | \( 1 - 3.52e35T + 1.01e71T^{2} \) |
| 89 | \( 1 - 9.72e35T + 1.34e72T^{2} \) |
| 97 | \( 1 - 5.14e36T + 3.24e73T^{2} \) |
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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
−11.22014308618343784578898871291, −9.315317849815251194355491775740, −9.100678349055553651466598793283, −7.11367186286882196869707634457, −6.22084889704499726169501892291, −5.17881124152593364690632714689, −3.82542159215651987814741916515, −3.20733077565271090771995930476, −2.34766874101216608176359743725, −0.45507999754599472885203306013,
0.45507999754599472885203306013, 2.34766874101216608176359743725, 3.20733077565271090771995930476, 3.82542159215651987814741916515, 5.17881124152593364690632714689, 6.22084889704499726169501892291, 7.11367186286882196869707634457, 9.100678349055553651466598793283, 9.315317849815251194355491775740, 11.22014308618343784578898871291
