| L(s) = 1 | + (−2.29 − 1.93i)3-s + (−8.17 − 4.71i)5-s + (1.49 + 8.87i)9-s + (−13.4 + 7.76i)11-s − 6.22·13-s + (9.58 + 26.6i)15-s + (−2.89 + 1.67i)17-s + (4.46 − 7.73i)19-s + (−16.3 − 9.43i)23-s + (32.0 + 55.4i)25-s + (13.7 − 23.2i)27-s − 41.0i·29-s + (−2.35 − 4.07i)31-s + (45.8 + 8.26i)33-s + (21.9 − 37.9i)37-s + ⋯ |
| L(s) = 1 | + (−0.763 − 0.645i)3-s + (−1.63 − 0.943i)5-s + (0.166 + 0.986i)9-s + (−1.22 + 0.705i)11-s − 0.479·13-s + (0.638 + 1.77i)15-s + (−0.170 + 0.0983i)17-s + (0.235 − 0.407i)19-s + (−0.710 − 0.410i)23-s + (1.28 + 2.21i)25-s + (0.509 − 0.860i)27-s − 1.41i·29-s + (−0.0759 − 0.131i)31-s + (1.38 + 0.250i)33-s + (0.592 − 1.02i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4177644603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4177644603\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.29 + 1.93i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (8.17 + 4.71i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (13.4 - 7.76i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 6.22T + 169T^{2} \) |
| 17 | \( 1 + (2.89 - 1.67i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-4.46 + 7.73i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (16.3 + 9.43i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 41.0iT - 841T^{2} \) |
| 31 | \( 1 + (2.35 + 4.07i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-21.9 + 37.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 46.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 47.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-21.1 - 12.1i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (1.86 - 1.07i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (11.0 - 6.38i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (9.94 - 17.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-40.4 - 70.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 65.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (50.2 + 87.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-10.7 + 18.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 49.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (40.3 + 23.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 36.3T + 9.40e3T^{2} \) |
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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
−10.79842550596506270492952146164, −9.702497482576476442079538736248, −8.422308917421451366059514959234, −7.69186394857111164123693400572, −7.29835124164100285158037869531, −5.85837512437244777286494179712, −4.81263380671370097308248402040, −4.23374278022280987602006893509, −2.43564423404836450954439769151, −0.67945797061032306952437678706,
0.29826804520407752400269144039, 2.94853836072121457602667564588, 3.74826802371942141441974347063, 4.78174217834835251516610384346, 5.81435698297375464346557579031, 6.96584100671586879226384062256, 7.68823222054181829012095331913, 8.577550497621920572111485991520, 9.896682058904556891956795043470, 10.76975039428865028479146801898
