| L(s) = 1 | + (−1.81 − 2.09i)2-s + (1.42 + 0.914i)3-s + (−0.814 + 5.66i)4-s + (0.415 − 0.909i)5-s + (−0.669 − 4.65i)6-s + (3.21 + 0.944i)7-s + (8.69 − 5.59i)8-s + (−0.0567 − 0.124i)9-s + (−2.66 + 0.782i)10-s + (−1.03 + 1.19i)11-s + (−6.33 + 7.31i)12-s + (1.00 − 0.295i)13-s + (−3.87 − 8.47i)14-s + (1.42 − 0.914i)15-s + (−16.5 − 4.87i)16-s + (−0.503 − 3.50i)17-s + ⋯ |
| L(s) = 1 | + (−1.28 − 1.48i)2-s + (0.821 + 0.528i)3-s + (−0.407 + 2.83i)4-s + (0.185 − 0.406i)5-s + (−0.273 − 1.90i)6-s + (1.21 + 0.357i)7-s + (3.07 − 1.97i)8-s + (−0.0189 − 0.0413i)9-s + (−0.843 + 0.247i)10-s + (−0.311 + 0.359i)11-s + (−1.83 + 2.11i)12-s + (0.279 − 0.0819i)13-s + (−1.03 − 2.26i)14-s + (0.367 − 0.236i)15-s + (−4.14 − 1.21i)16-s + (−0.122 − 0.849i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.676306 - 0.400666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.676306 - 0.400666i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (-0.292 - 4.78i)T \) |
| good | 2 | \( 1 + (1.81 + 2.09i)T + (-0.284 + 1.97i)T^{2} \) |
| 3 | \( 1 + (-1.42 - 0.914i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (-3.21 - 0.944i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (1.03 - 1.19i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 0.295i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.503 + 3.50i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.243 - 1.69i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.0116 + 0.0807i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (1.70 - 1.09i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.66 + 5.82i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (4.70 - 10.2i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.841 + 0.541i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 + (8.88 + 2.60i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-0.489 + 0.143i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (6.69 - 4.30i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-3.09 - 3.56i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (7.56 + 8.73i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.624 - 4.34i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-2.19 + 0.644i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.50 + 3.28i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-9.47 - 6.08i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.90 + 6.36i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−13.09739719180749064926579327094, −11.94578564234809723902227276736, −11.23435879050918747678165263331, −10.03680561963807793347484771304, −9.237462370846255503326090630893, −8.470536604808618166290778582574, −7.65031830641640053726637802369, −4.67372961006877113062806622540, −3.23969392023979154385911314681, −1.79948423938142276649738380563,
1.78846026398664394889315739489, 4.99337807952598412199489058356, 6.40096190844521297157992436405, 7.50113404298913848370359693251, 8.234707213492056816188954216488, 8.875617571870130532916664593614, 10.39282931370146684062744157543, 11.05272036915433130057785302240, 13.40544926050204588464508152590, 14.20492117708331599871300128452
