| L(s) = 1 | + (−1.34 − 0.441i)2-s + (−0.157 − 0.288i)3-s + (1.61 + 1.18i)4-s + (0.136 + 2.23i)5-s + (0.0843 + 0.457i)6-s + (2.06 + 2.75i)7-s + (−1.63 − 2.30i)8-s + (1.56 − 2.43i)9-s + (0.801 − 3.05i)10-s + (−4.78 + 2.18i)11-s + (0.0886 − 0.652i)12-s + (4.22 + 3.16i)13-s + (−1.55 − 4.61i)14-s + (0.623 − 0.391i)15-s + (1.18 + 3.82i)16-s + (−4.79 + 0.342i)17-s + ⋯ |
| L(s) = 1 | + (−0.950 − 0.312i)2-s + (−0.0910 − 0.166i)3-s + (0.805 + 0.593i)4-s + (0.0611 + 0.998i)5-s + (0.0344 + 0.186i)6-s + (0.779 + 1.04i)7-s + (−0.579 − 0.814i)8-s + (0.521 − 0.810i)9-s + (0.253 − 0.967i)10-s + (−1.44 + 0.658i)11-s + (0.0255 − 0.188i)12-s + (1.17 + 0.877i)13-s + (−0.415 − 1.23i)14-s + (0.160 − 0.101i)15-s + (0.296 + 0.955i)16-s + (−1.16 + 0.0831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.619110 + 0.534534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.619110 + 0.534534i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 + 0.441i)T \) |
| 5 | \( 1 + (-0.136 - 2.23i)T \) |
| 23 | \( 1 + (0.722 - 4.74i)T \) |
| good | 3 | \( 1 + (0.157 + 0.288i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-2.06 - 2.75i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (4.78 - 2.18i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-4.22 - 3.16i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (4.79 - 0.342i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (3.72 + 4.29i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.65 - 3.16i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.613 + 2.08i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.43 - 6.58i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (-7.31 + 4.69i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.00 - 5.50i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (0.274 - 0.274i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.31 - 5.76i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (-1.02 - 7.12i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (5.30 - 1.55i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (2.49 + 6.68i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (0.758 + 0.346i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-4.31 - 0.308i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.414 - 2.87i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.562 - 2.58i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-2.32 + 7.93i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-4.97 - 1.08i)T + (88.2 + 40.2i)T^{2} \) |
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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
−11.13161827731438643043420586391, −10.51473832767502810663852181353, −9.373398503303006455143464577152, −8.710214612463968112621246359239, −7.66647804100934185821844153977, −6.79229421387801281108302911204, −6.00205885731780518458232975578, −4.31414991955956181791076630401, −2.76406449593061721845079692341, −1.86745548813684024697994350338,
0.68977876873772464381379963174, 2.13974483833961434352723796227, 4.22360663244469230655765052019, 5.21654084839043116155838231146, 6.18028574773613540571180136377, 7.67019618142587268697825833052, 8.127230619528964935097189694015, 8.746343263061456499736703176770, 10.23300328198347553086869232613, 10.64052833932728359881785797314
