| L(s) = 1 | + (−1.03 + 0.963i)2-s + (−0.263 − 0.184i)3-s + (0.142 − 1.99i)4-s + (−3.33 − 0.291i)5-s + (0.450 − 0.0629i)6-s + (2.88 + 1.66i)7-s + (1.77 + 2.20i)8-s + (−0.990 − 2.72i)9-s + (3.72 − 2.90i)10-s + (−0.0699 + 0.261i)11-s + (−0.405 + 0.498i)12-s + (2.96 + 4.22i)13-s + (−4.58 + 1.05i)14-s + (0.822 + 0.690i)15-s + (−3.95 − 0.568i)16-s + (5.53 + 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.731 + 0.681i)2-s + (−0.151 − 0.106i)3-s + (0.0712 − 0.997i)4-s + (−1.48 − 0.130i)5-s + (0.183 − 0.0256i)6-s + (1.08 + 0.629i)7-s + (0.627 + 0.778i)8-s + (−0.330 − 0.907i)9-s + (1.17 − 0.919i)10-s + (−0.0210 + 0.0787i)11-s + (−0.116 + 0.143i)12-s + (0.821 + 1.17i)13-s + (−1.22 + 0.282i)14-s + (0.212 + 0.178i)15-s + (−0.989 − 0.142i)16-s + (1.34 + 0.488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.647432 + 0.381894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647432 + 0.381894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.03 - 0.963i)T \) |
| 19 | \( 1 + (-3.91 - 1.91i)T \) |
| good | 3 | \( 1 + (0.263 + 0.184i)T + (1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (3.33 + 0.291i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (-2.88 - 1.66i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0699 - 0.261i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.96 - 4.22i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-5.53 - 2.01i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.724 + 0.863i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.68 - 5.75i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-2.08 + 3.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.13 - 5.13i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.65 + 0.468i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (8.38 + 0.733i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-1.31 + 0.479i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.172 - 1.96i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (3.67 + 7.88i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 0.906i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-3.01 + 6.46i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-6.26 - 7.46i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (4.62 - 0.814i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.308 - 1.74i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.42 + 12.7i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.0559 - 0.00986i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (5.21 + 1.89i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.49012303756151605205491361455, −11.33933253170982413551908859708, −9.771536523674475241396206796389, −8.687100515266536221844223236230, −8.159237067237097115189221514247, −7.27622863619419807316550282146, −6.11559089582944494506237616590, −4.99345417295562049981874001064, −3.68221771225378162185858062937, −1.28727794351078240383178109019,
0.909349484498338907813620300847, 3.02806636330398791026224497150, 4.07920892367366731673000448171, 5.27137834064178782446356769895, 7.41608466303457642211377623858, 7.84362531977685408501698856680, 8.410679721515158136487187090992, 9.948864831617854702407661320572, 10.91390387029816129305126193021, 11.32322502273019415782102281109
