| L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.345 − 0.289i)3-s + (−0.939 + 0.342i)4-s + (1.87 + 0.681i)5-s + (−0.345 − 0.289i)6-s + (−1.54 + 2.67i)7-s + (0.5 + 0.866i)8-s + (−0.485 + 2.75i)9-s + (0.346 − 1.96i)10-s + (−0.5 − 0.866i)11-s + (−0.225 + 0.390i)12-s + (4.93 + 4.13i)13-s + (2.90 + 1.05i)14-s + (0.843 − 0.307i)15-s + (0.766 − 0.642i)16-s + (0.389 + 2.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.199 − 0.167i)3-s + (−0.469 + 0.171i)4-s + (0.837 + 0.304i)5-s + (−0.140 − 0.118i)6-s + (−0.583 + 1.01i)7-s + (0.176 + 0.306i)8-s + (−0.161 + 0.918i)9-s + (0.109 − 0.620i)10-s + (−0.150 − 0.261i)11-s + (−0.0650 + 0.112i)12-s + (1.36 + 1.14i)13-s + (0.775 + 0.282i)14-s + (0.217 − 0.0793i)15-s + (0.191 − 0.160i)16-s + (0.0944 + 0.535i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39407 + 0.144337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39407 + 0.144337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-1.42 + 4.11i)T \) |
| good | 3 | \( 1 + (-0.345 + 0.289i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-1.87 - 0.681i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.54 - 2.67i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-4.93 - 4.13i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.389 - 2.20i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (6.08 - 2.21i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.341 - 1.93i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.49 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.00T + 37T^{2} \) |
| 41 | \( 1 + (-7.98 + 6.70i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-11.3 - 4.14i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.00 + 11.3i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (6.22 - 2.26i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.714 - 4.05i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (12.1 - 4.43i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.800 + 4.54i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.88 + 1.77i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.73 + 7.32i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (9.31 - 7.81i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.739 + 1.28i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.67 + 2.24i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.480 + 2.72i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.17074612194524937342802933583, −10.41171065055465662132519943867, −9.333449866170078791906607770551, −8.849449499137733734740354026053, −7.74411296961170933809883929611, −6.24287352898449206294359167135, −5.67400020336605652177873625162, −4.11248080446398950864811973451, −2.70640755213749343077647472325, −1.88642023363763532095168279082,
1.00487485321539627319474490211, 3.24911825626476111231520532716, 4.27700763653247661251975099578, 5.84127689763572718545874296526, 6.21397797841356796843121526782, 7.50855641152734606045576434400, 8.383181076735187838952585661685, 9.472776561560704993457634183518, 9.981070465873421163522555327933, 10.84108933297811548417252837526
