| L(s) = 1 | + (0.959 + 0.281i)2-s + (1.48 − 1.71i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (1.90 − 1.22i)6-s + (−0.119 − 0.261i)7-s + (0.654 + 0.755i)8-s + (−0.302 − 2.10i)9-s + (−0.415 + 0.909i)10-s + (−1.41 + 0.415i)11-s + (2.17 − 0.637i)12-s + (0.722 − 1.58i)13-s + (−0.0408 − 0.284i)14-s + (1.48 + 1.71i)15-s + (0.415 + 0.909i)16-s + (−4.20 + 2.69i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (0.855 − 0.987i)3-s + (0.420 + 0.270i)4-s + (−0.0636 + 0.442i)5-s + (0.777 − 0.499i)6-s + (−0.0450 − 0.0987i)7-s + (0.231 + 0.267i)8-s + (−0.100 − 0.700i)9-s + (−0.131 + 0.287i)10-s + (−0.427 + 0.125i)11-s + (0.626 − 0.184i)12-s + (0.200 − 0.438i)13-s + (−0.0109 − 0.0759i)14-s + (0.382 + 0.441i)15-s + (0.103 + 0.227i)16-s + (−1.01 + 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12697 - 0.317839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12697 - 0.317839i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-3.72 + 3.01i)T \) |
| good | 3 | \( 1 + (-1.48 + 1.71i)T + (-0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (0.119 + 0.261i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (1.41 - 0.415i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.722 + 1.58i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (4.20 - 2.69i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (4.28 + 2.75i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.22 - 2.71i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.0279 + 0.0322i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.623 - 4.33i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.14 - 7.97i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.56 - 2.96i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 4.95T + 47T^{2} \) |
| 53 | \( 1 + (0.805 + 1.76i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.26 + 4.96i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (6.42 + 7.41i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-6.47 - 1.90i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (3.42 + 1.00i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (9.83 + 6.31i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.63 + 12.3i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.38 - 9.65i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-2.15 + 2.49i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.689 - 4.79i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−12.72377595630955217849629289262, −11.28876660609936489183520160050, −10.48016205602186491584076023719, −8.844956062165359908420503643527, −8.040160097115847317019737271418, −7.04584626784092122604690724937, −6.31665978456193731003788801971, −4.70752579782701338641639557335, −3.19865778120746263292157753158, −2.11471909547319124959742831945,
2.37540101058078123445749107209, 3.74947768417077342331600734006, 4.55164610791908854774180115223, 5.76790966012903891527725531665, 7.25119656776854210613086200591, 8.652634241346314944199650363341, 9.258892606623785886878754314534, 10.36585923044426133144937833483, 11.23661585787738611665482289465, 12.36460957665625754115497339066
