| L(s) = 1 | − 0.618i·2-s + i·3-s + 1.61·4-s + 0.618·6-s + 1.61i·7-s − 2.23i·8-s + 2·9-s − 0.763·11-s + 1.61i·12-s − 4.85i·13-s + 1.00·14-s + 1.85·16-s + 0.763i·17-s − 1.23i·18-s + 5.85·19-s + ⋯ |
| L(s) = 1 | − 0.437i·2-s + 0.577i·3-s + 0.809·4-s + 0.252·6-s + 0.611i·7-s − 0.790i·8-s + 0.666·9-s − 0.230·11-s + 0.467i·12-s − 1.34i·13-s + 0.267·14-s + 0.463·16-s + 0.185i·17-s − 0.291i·18-s + 1.34·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94765\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.618iT - 2T^{2} \) |
| 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 1.61iT - 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.85iT - 13T^{2} \) |
| 17 | \( 1 - 0.763iT - 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 - 8.23iT - 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 4.23iT - 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 1.85iT - 43T^{2} \) |
| 47 | \( 1 - 1.61iT - 47T^{2} \) |
| 53 | \( 1 - 5.47iT - 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 + 9.23iT - 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.76iT - 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 2.85iT - 97T^{2} \) |
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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
−10.56955190629741239830137216016, −9.913190257641533391398029939563, −9.188234492487465265958063548697, −7.80228437069641996646572465208, −7.24151197158025580684696081866, −5.87283574654788301851021878817, −5.19128497517591427854368968062, −3.67893552820163325018789607638, −2.91673754300495098929424786061, −1.45905467638233249020922176924,
1.36663032069135782685070730924, 2.56898862532313712441395724986, 4.07597015596526848489318655256, 5.22431113126404089311311913072, 6.55920480133898467989065774296, 6.93351606624168514841384955977, 7.67257166294978340290769671102, 8.634784460440154073532331568500, 9.859704918522041322989772108011, 10.59380952454756027919053582077
