| L(s) = 1 | + (−1.35 + 0.419i)2-s + (−1.23 + 0.450i)3-s + (1.64 − 1.13i)4-s + (−0.503 + 2.85i)5-s + (1.48 − 1.12i)6-s + (−2.71 + 1.56i)7-s + (−1.74 + 2.22i)8-s + (−0.971 + 0.815i)9-s + (−0.517 − 4.06i)10-s + (3.30 + 1.90i)11-s + (−1.52 + 2.14i)12-s + (1.53 − 4.21i)13-s + (3.00 − 3.25i)14-s + (−0.661 − 3.75i)15-s + (1.43 − 3.73i)16-s + (0.0599 + 0.0503i)17-s + ⋯ |
| L(s) = 1 | + (−0.954 + 0.296i)2-s + (−0.713 + 0.259i)3-s + (0.823 − 0.566i)4-s + (−0.224 + 1.27i)5-s + (0.604 − 0.460i)6-s + (−1.02 + 0.591i)7-s + (−0.618 + 0.785i)8-s + (−0.323 + 0.271i)9-s + (−0.163 − 1.28i)10-s + (0.996 + 0.575i)11-s + (−0.440 + 0.618i)12-s + (0.425 − 1.16i)13-s + (0.802 − 0.869i)14-s + (−0.170 − 0.969i)15-s + (0.357 − 0.933i)16-s + (0.0145 + 0.0122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.215626 + 0.361364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.215626 + 0.361364i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 - 0.419i)T \) |
| 19 | \( 1 + (-3.51 - 2.57i)T \) |
| good | 3 | \( 1 + (1.23 - 0.450i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (0.503 - 2.85i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.71 - 1.56i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.30 - 1.90i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 4.21i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.0599 - 0.0503i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (2.21 - 0.390i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.332 - 0.395i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.35 + 2.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 + (-0.203 - 0.558i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-10.9 - 1.92i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.45 - 1.73i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.929 + 0.163i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (6.93 + 5.82i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.585 + 3.32i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.61 - 1.35i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.503 - 2.85i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-13.7 + 4.99i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (6.53 - 2.37i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.69 + 2.71i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.01 + 2.79i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (6.08 - 7.25i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−15.23326298701659553141164975853, −14.17585319214002619906417297979, −12.26487676475928265371280207264, −11.29972626543549629454809825798, −10.37370556363898611965414344698, −9.500076284429431979485103932226, −7.897457793431825257186158211935, −6.57463784299572162076981288904, −5.79505211665235865660728865060, −3.02584958772916728467098761434,
0.838582670727582971617464584926, 3.83925275746258021642033186789, 6.04412645136327867100078230098, 7.11326083955179187258484559793, 8.865732904210133603960565945621, 9.340433378691647588155948397244, 10.98500357952848251667834665671, 11.91711353844352249156367432894, 12.61286856714346133671928937484, 13.93150762415654381748869751186
