| L(s) = 1 | + (−0.665 − 1.24i)2-s + (0.987 + 0.156i)3-s + (−1.11 + 1.66i)4-s + (−0.495 + 2.18i)5-s + (−0.461 − 1.33i)6-s + (−0.602 + 0.602i)7-s + (2.81 + 0.288i)8-s + (0.951 + 0.309i)9-s + (3.05 − 0.832i)10-s + (−5.05 + 1.64i)11-s + (−1.36 + 1.46i)12-s + (1.03 + 2.03i)13-s + (1.15 + 0.351i)14-s + (−0.830 + 2.07i)15-s + (−1.51 − 3.70i)16-s + (0.799 + 5.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.470 − 0.882i)2-s + (0.570 + 0.0903i)3-s + (−0.557 + 0.830i)4-s + (−0.221 + 0.975i)5-s + (−0.188 − 0.545i)6-s + (−0.227 + 0.227i)7-s + (0.994 + 0.101i)8-s + (0.317 + 0.103i)9-s + (0.964 − 0.263i)10-s + (−1.52 + 0.495i)11-s + (−0.393 + 0.422i)12-s + (0.287 + 0.564i)13-s + (0.308 + 0.0938i)14-s + (−0.214 + 0.536i)15-s + (−0.377 − 0.925i)16-s + (0.193 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.901450 + 0.356164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.901450 + 0.356164i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.665 + 1.24i)T \) |
| 3 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (0.495 - 2.18i)T \) |
| good | 7 | \( 1 + (0.602 - 0.602i)T - 7iT^{2} \) |
| 11 | \( 1 + (5.05 - 1.64i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 2.03i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.799 - 5.04i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-6.89 - 5.01i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.489 + 0.961i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (4.22 + 5.81i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.29 + 5.91i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.750 + 0.382i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-2.32 + 7.15i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-4.73 - 4.73i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.212 + 1.34i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (1.49 - 9.46i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (0.145 - 0.448i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.80 + 8.61i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-2.40 + 0.381i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (1.56 + 2.15i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (11.2 + 5.73i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-1.31 + 0.956i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.566 - 3.57i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-8.73 + 2.83i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-13.8 - 2.19i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.74352667989457213810825967632, −10.71227524233262993934177682594, −10.09152815356065250317735024225, −9.296054955030881606144918567860, −7.81458833914290495062836756540, −7.69038074159284632336252731189, −5.91674801077962056557384134820, −4.20285359091675630362826268489, −3.15788265397059614094595375340, −2.12283284083899605513489021895,
0.810156826064321964002296120161, 3.11225450546868621135234188407, 4.91049312461422019673098856536, 5.46808644003515778479615125974, 7.15356974357169109273255530916, 7.78475038901738309125352832323, 8.688702640169064067248652986864, 9.441746754399083200354939632465, 10.36854332123686195858442723074, 11.55422240061417168142108034452
