| L(s) = 1 | + 2.15i·3-s − i·5-s + 0.557·7-s − 1.63·9-s − 3.17·11-s − 2.16·13-s + 2.15·15-s − 2.43i·17-s + 4.35·19-s + 1.20i·21-s + (−4.29 + 2.13i)23-s − 25-s + 2.94i·27-s + 0.0880·29-s + 2.25i·31-s + ⋯ |
| L(s) = 1 | + 1.24i·3-s − 0.447i·5-s + 0.210·7-s − 0.543·9-s − 0.957·11-s − 0.601·13-s + 0.555·15-s − 0.591i·17-s + 0.998·19-s + 0.261i·21-s + (−0.895 + 0.444i)23-s − 0.200·25-s + 0.566i·27-s + 0.0163·29-s + 0.404i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2556700387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2556700387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (4.29 - 2.13i)T \) |
| good | 3 | \( 1 - 2.15iT - 3T^{2} \) |
| 7 | \( 1 - 0.557T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 + 2.43iT - 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 29 | \( 1 - 0.0880T + 29T^{2} \) |
| 31 | \( 1 - 2.25iT - 31T^{2} \) |
| 37 | \( 1 - 2.11iT - 37T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 + 0.0507T + 43T^{2} \) |
| 47 | \( 1 + 9.01iT - 47T^{2} \) |
| 53 | \( 1 + 8.53iT - 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 + 9.91iT - 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 + 12.5iT - 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 7.99T + 79T^{2} \) |
| 83 | \( 1 + 7.03T + 83T^{2} \) |
| 89 | \( 1 + 8.31iT - 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−8.395919164934154704678611387146, −7.68216163000561370649870289601, −6.94231262084196173908975271332, −5.70307994833570973544899780798, −5.06401721895713199208707838353, −4.70802470210016766487241977246, −3.66780750505198218222843087931, −2.94743296288345183135810699701, −1.71877168724019088628083377729, −0.07278704254141279941588767040,
1.32007003536084118618707296884, 2.26049276542335917311735157462, 2.95773227672531424678175897335, 4.13478534311128740133774423328, 5.14397920310068569812008302246, 5.93231834684281658340737617811, 6.63269911874252321270407115368, 7.39090319686647537380326067396, 7.84940480120008346106759726117, 8.388159427747273855972264161743
