| L(s) = 1 | + (1.88 − 0.838i)2-s + (0.927 − 0.117i)3-s + (1.50 − 1.67i)4-s + (−0.800 + 2.86i)5-s + (1.64 − 0.998i)6-s + (−1.17 − 3.17i)7-s + (0.161 − 0.496i)8-s + (−2.06 + 0.528i)9-s + (0.896 + 6.07i)10-s + (2.18 − 2.87i)11-s + (1.20 − 1.72i)12-s + (−3.92 + 1.36i)13-s + (−4.88 − 4.98i)14-s + (−0.406 + 2.75i)15-s + (0.358 + 3.41i)16-s + (0.211 − 0.0357i)17-s + ⋯ |
| L(s) = 1 | + (1.33 − 0.593i)2-s + (0.535 − 0.0676i)3-s + (0.753 − 0.837i)4-s + (−0.357 + 1.28i)5-s + (0.672 − 0.407i)6-s + (−0.445 − 1.19i)7-s + (0.0570 − 0.175i)8-s + (−0.686 + 0.176i)9-s + (0.283 + 1.92i)10-s + (0.657 − 0.866i)11-s + (0.346 − 0.499i)12-s + (−1.08 + 0.379i)13-s + (−1.30 − 1.33i)14-s + (−0.104 + 0.710i)15-s + (0.0896 + 0.852i)16-s + (0.0512 − 0.00866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.05146 - 0.545233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05146 - 0.545233i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (1.11 - 12.2i)T \) |
| good | 2 | \( 1 + (-1.88 + 0.838i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.927 + 0.117i)T + (2.90 - 0.746i)T^{2} \) |
| 5 | \( 1 + (0.800 - 2.86i)T + (-4.27 - 2.59i)T^{2} \) |
| 7 | \( 1 + (1.17 + 3.17i)T + (-5.29 + 4.57i)T^{2} \) |
| 11 | \( 1 + (-2.18 + 2.87i)T + (-2.95 - 10.5i)T^{2} \) |
| 13 | \( 1 + (3.92 - 1.36i)T + (10.1 - 8.07i)T^{2} \) |
| 17 | \( 1 + (-0.211 + 0.0357i)T + (16.0 - 5.59i)T^{2} \) |
| 19 | \( 1 + (1.76 + 5.43i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-7.27 - 1.54i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-2.67 - 2.51i)T + (1.82 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-2.59 - 1.28i)T + (18.7 + 24.6i)T^{2} \) |
| 37 | \( 1 + (-6.79 - 5.86i)T + (5.40 + 36.6i)T^{2} \) |
| 41 | \( 1 + (6.09 + 3.35i)T + (21.9 + 34.6i)T^{2} \) |
| 43 | \( 1 + (1.65 - 4.44i)T + (-32.5 - 28.0i)T^{2} \) |
| 47 | \( 1 + (0.182 + 8.71i)T + (-46.9 + 1.96i)T^{2} \) |
| 53 | \( 1 + (-3.72 + 7.91i)T + (-33.7 - 40.8i)T^{2} \) |
| 59 | \( 1 + (10.0 - 7.30i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.98 - 4.72i)T + (-42.6 + 43.5i)T^{2} \) |
| 67 | \( 1 + (8.40 + 2.15i)T + (58.7 + 32.2i)T^{2} \) |
| 71 | \( 1 + (9.50 + 1.60i)T + (67.0 + 23.3i)T^{2} \) |
| 73 | \( 1 + (-1.91 + 2.31i)T + (-13.6 - 71.7i)T^{2} \) |
| 79 | \( 1 + (-0.550 + 8.74i)T + (-78.3 - 9.90i)T^{2} \) |
| 83 | \( 1 + (-3.63 + 5.72i)T + (-35.3 - 75.1i)T^{2} \) |
| 89 | \( 1 + (-2.22 - 0.0932i)T + (88.6 + 7.44i)T^{2} \) |
| 97 | \( 1 + (2.94 - 3.00i)T + (-2.03 - 96.9i)T^{2} \) |
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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
−13.43613634341447815434375788527, −11.81557745796018622378210088642, −11.18066764482532021176128798121, −10.36706021027064985243769955038, −8.819478106174686804549269474929, −7.26240618766041078628425602351, −6.46381055692205779578778890713, −4.74968460698917306136975779130, −3.40057294706135153343235985207, −2.82546716787077270796392841270,
2.80203680377937206910157689497, 4.29864711551662014458027532483, 5.25408535962157316617407003563, 6.26386683801972742184501213262, 7.76932780613354960026379795827, 8.905137778864423092750252100479, 9.641214852205932051330455234071, 11.88337965527268604431532470848, 12.39757896046710468336265586102, 12.92385596242006027488546492808
