| L(s) = 1 | − 1.09e5i·2-s − 1.96e7i·3-s − 3.34e9·4-s − 2.14e12·6-s − 7.24e12i·7-s − 5.73e14i·8-s + 5.17e15·9-s + 1.80e17·11-s + 6.55e16i·12-s + 1.38e18i·13-s − 7.91e17·14-s − 9.13e19·16-s + 6.46e19i·17-s − 5.65e20i·18-s − 1.46e21·19-s + ⋯ |
| L(s) = 1 | − 1.17i·2-s − 0.263i·3-s − 0.389·4-s − 0.310·6-s − 0.0823i·7-s − 0.719i·8-s + 0.930·9-s + 1.18·11-s + 0.102i·12-s + 0.576i·13-s − 0.0970·14-s − 1.23·16-s + 0.322i·17-s − 1.09i·18-s − 1.16·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(2.816646711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.816646711\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.09e5iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 1.96e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 + 7.24e12iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 1.80e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 1.38e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 - 6.46e19iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 1.46e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 7.51e21iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 3.42e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 1.13e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 2.61e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 1.74e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.37e27iT - 8.02e53T^{2} \) |
| 47 | \( 1 - 3.57e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 3.47e27iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 1.06e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 3.27e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 5.06e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 + 2.35e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 2.40e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 2.85e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 1.04e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 - 2.05e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 7.08e32iT - 3.65e65T^{2} \) |
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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
−10.89078533779913091779601371657, −9.896104117323418306734718864204, −8.853294902110913202859767816201, −7.16672155848567451423643806239, −6.33264237485199331116334398810, −4.35804126719975718391199410032, −3.72981188868636099513220822895, −2.23737780065675457844000144119, −1.52407746609715373551509999976, −0.59025419090698080854751896361,
0.994108593744579810490600922696, 2.32290301491358780729913937404, 3.93280592316741564424580495538, 4.93500225587971135119260964988, 6.22739031645599782283648899491, 6.96385934461357244233700959769, 8.129538057592796437906938672350, 9.206859818634557665354732057551, 10.49603063981217424902635886538, 11.79661534327485759535597206217
