| L(s) = 1 | + (−0.430 − 1.34i)2-s + (0.755 + 0.654i)3-s + (−1.62 + 1.16i)4-s + (−0.320 − 0.0460i)5-s + (0.556 − 1.30i)6-s + (0.756 + 1.65i)7-s + (2.26 + 1.69i)8-s + (0.142 + 0.989i)9-s + (0.0759 + 0.451i)10-s + (5.31 − 1.55i)11-s + (−1.99 − 0.189i)12-s + (−0.558 + 1.22i)13-s + (1.90 − 1.73i)14-s + (−0.211 − 0.244i)15-s + (1.30 − 3.78i)16-s + (0.179 + 0.279i)17-s + ⋯ |
| L(s) = 1 | + (−0.304 − 0.952i)2-s + (0.436 + 0.378i)3-s + (−0.814 + 0.580i)4-s + (−0.143 − 0.0205i)5-s + (0.227 − 0.530i)6-s + (0.285 + 0.626i)7-s + (0.800 + 0.598i)8-s + (0.0474 + 0.329i)9-s + (0.0240 + 0.142i)10-s + (1.60 − 0.470i)11-s + (−0.574 − 0.0547i)12-s + (−0.154 + 0.339i)13-s + (0.509 − 0.463i)14-s + (−0.0547 − 0.0631i)15-s + (0.326 − 0.945i)16-s + (0.0435 + 0.0676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23262 - 0.198696i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23262 - 0.198696i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.430 + 1.34i)T \) |
| 3 | \( 1 + (-0.755 - 0.654i)T \) |
| 23 | \( 1 + (-4.71 + 0.888i)T \) |
| good | 5 | \( 1 + (0.320 + 0.0460i)T + (4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.756 - 1.65i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-5.31 + 1.55i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.558 - 1.22i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.179 - 0.279i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.77 - 2.42i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.0345 - 0.0222i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.59 - 3.11i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.31 - 0.763i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.676 + 4.70i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.856 + 0.988i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 6.53iT - 47T^{2} \) |
| 53 | \( 1 + (6.61 - 3.02i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 0.561i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (3.49 - 3.02i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (15.5 + 4.57i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.46 - 8.41i)T + (-59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-11.6 - 7.49i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.32 + 2.89i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.04 - 7.23i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (9.43 + 8.17i)T + (12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-4.13 - 0.595i)T + (93.0 + 27.3i)T^{2} \) |
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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
−11.80422106659059617801270865452, −10.98811512964357770818507440044, −9.842343565471494793090914743067, −9.039555686274970219284385066347, −8.479206831150042037433815089809, −7.17255585937000445582605965599, −5.50711813935764468275107208435, −4.18193081291588301744090824212, −3.23721279671136069329897150875, −1.66688454487239017759342917320,
1.29890661513668178638476770700, 3.66764785115687086445919229161, 4.82657844350248276188418297572, 6.24801858737455455482076384697, 7.22695685597217540141729016452, 7.77392280089175315074086404260, 9.102685776707907919566440639519, 9.558265874626022627218363575244, 10.90200051441219155986541239919, 11.99218117103306794499758359314
