| L(s) = 1 | + (−1.70 + 0.292i)3-s + (−1 − i)7-s + (2.82 − i)9-s + 1.41i·11-s + (−1.41 + 1.41i)17-s − 4i·19-s + (2 + 1.41i)21-s + (2.82 + 2.82i)23-s + (−4.53 + 2.53i)27-s − 7.07·29-s + 2·31-s + (−0.414 − 2.41i)33-s + (−6 − 6i)37-s − 5.65i·41-s + (6 − 6i)43-s + ⋯ |
| L(s) = 1 | + (−0.985 + 0.169i)3-s + (−0.377 − 0.377i)7-s + (0.942 − 0.333i)9-s + 0.426i·11-s + (−0.342 + 0.342i)17-s − 0.917i·19-s + (0.436 + 0.308i)21-s + (0.589 + 0.589i)23-s + (−0.872 + 0.487i)27-s − 1.31·29-s + 0.359·31-s + (−0.0721 − 0.420i)33-s + (−0.986 − 0.986i)37-s − 0.883i·41-s + (0.914 − 0.914i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5634874335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5634874335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.292i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (6 + 6i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-6 + 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-2.82 - 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (4 + 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−9.473413315592910463185919784607, −8.985943537736336382726136420825, −7.48386243272368304393158831087, −7.05396175211584730588091746628, −6.09672682440284527550956039905, −5.26771250884619389836909941152, −4.37813408933350419936189199090, −3.46016032193849523947999762309, −1.85912937257767425104579989893, −0.29118984894164705980880808373,
1.32378838970791182785476575744, 2.77733370164046151994765170182, 4.03916316560390462091312701784, 5.04128471302963264032310056062, 5.87874776283716176884648340049, 6.51857339768752442214269681994, 7.40725929661035818849937305248, 8.330155677953828421286135520151, 9.325073647318396619389891506950, 10.05113670662118048210873635550
