| L(s) = 1 | + (−0.625 + 1.26i)2-s + (−0.178 + 0.489i)3-s + (−1.21 − 1.58i)4-s + (3.55 − 0.626i)5-s + (−0.509 − 0.532i)6-s + (−1.81 + 3.13i)7-s + (2.77 − 0.553i)8-s + (2.09 + 1.75i)9-s + (−1.42 + 4.89i)10-s + (−0.380 + 0.219i)11-s + (0.994 − 0.313i)12-s + (−1.07 − 2.95i)13-s + (−2.84 − 4.25i)14-s + (−0.326 + 1.85i)15-s + (−1.03 + 3.86i)16-s + (−5.14 + 4.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.442 + 0.896i)2-s + (−0.102 + 0.282i)3-s + (−0.609 − 0.793i)4-s + (1.58 − 0.280i)5-s + (−0.208 − 0.217i)6-s + (−0.684 + 1.18i)7-s + (0.980 − 0.195i)8-s + (0.696 + 0.584i)9-s + (−0.451 + 1.54i)10-s + (−0.114 + 0.0661i)11-s + (0.286 − 0.0905i)12-s + (−0.298 − 0.820i)13-s + (−0.760 − 1.13i)14-s + (−0.0843 + 0.478i)15-s + (−0.258 + 0.966i)16-s + (−1.24 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.735574 + 0.684944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.735574 + 0.684944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.625 - 1.26i)T \) |
| 19 | \( 1 + (-4.34 - 0.341i)T \) |
| good | 3 | \( 1 + (0.178 - 0.489i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-3.55 + 0.626i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.81 - 3.13i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.380 - 0.219i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.07 + 2.95i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (5.14 - 4.31i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.958 + 5.43i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.67 + 1.99i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 4.49i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.95iT - 37T^{2} \) |
| 41 | \( 1 + (-0.621 - 0.226i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.376 - 0.0663i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.54 + 2.97i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (13.0 + 2.30i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (8.50 + 10.1i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-5.02 - 0.885i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.576 + 0.687i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.20 - 6.83i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.14 - 1.14i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (6.23 + 2.26i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.47 - 3.15i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.76 + 1.00i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (3.66 - 3.07i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.15991625128996906432116636550, −12.77917032427069496511068209144, −10.70916500452942342631351301527, −9.849479955211933788466029145598, −9.270067975578250126327401712914, −8.168444087379111713174150088648, −6.54705678884153795927609062485, −5.77645415667424563765237237802, −4.85054004496194138239471314055, −2.16786953304182513179156510041,
1.43668445794291116000132072252, 3.08532091598378210333963206953, 4.71078828115455650521691265443, 6.58524177326440906013776474326, 7.27400451518404089962240608672, 9.312773463831623192917417015081, 9.638779193606996722629575814378, 10.53780229779606476752450831939, 11.65331920868745550220785423487, 12.88336262394071709667752758606
