| L(s) = 1 | + 1.61i·2-s + i·3-s − 0.618·4-s − 1.61·6-s − 0.618i·7-s + 2.23i·8-s + 2·9-s − 5.23·11-s − 0.618i·12-s + 1.85i·13-s + 1.00·14-s − 4.85·16-s + 5.23i·17-s + 3.23i·18-s − 0.854·19-s + ⋯ |
| L(s) = 1 | + 1.14i·2-s + 0.577i·3-s − 0.309·4-s − 0.660·6-s − 0.233i·7-s + 0.790i·8-s + 0.666·9-s − 1.57·11-s − 0.178i·12-s + 0.514i·13-s + 0.267·14-s − 1.21·16-s + 1.26i·17-s + 0.762i·18-s − 0.195·19-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
| \(L(1)\) |
\(\approx\) |
\(-1.35345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-1.35345i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.61iT - 2T^{2} \) |
| 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 + 0.618iT - 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 1.85iT - 13T^{2} \) |
| 17 | \( 1 - 5.23iT - 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 - 3.76iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 0.236iT - 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 + 4.85iT - 43T^{2} \) |
| 47 | \( 1 + 0.618iT - 47T^{2} \) |
| 53 | \( 1 + 3.47iT - 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 4.76iT - 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 + 6.23iT - 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 3.85iT - 97T^{2} \) |
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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
−10.66210096009775161821181164749, −10.32326723465647907036878606654, −9.126967961139171238911937529022, −8.191902905577469752040421952913, −7.49007802110720543897693615563, −6.65717171646911595805736848425, −5.59317323716215296973442642715, −4.85306140926171489508089612386, −3.73755592235147528671333569411, −2.11471709929814528298857042533,
0.72262107937754637892936497162, 2.25738003964280298841056092760, 2.92062907902905855595963599408, 4.36563928352683698584166063216, 5.44936856000700956702498363574, 6.77031415610353377858619929978, 7.48428064120398438121225915356, 8.459543414156430657438582226724, 9.699380988510999225722706772579, 10.29335548209582714484894782098
