| L(s) = 1 | + (0.330 + 0.943i)2-s + (−0.995 + 2.84i)3-s + (−0.781 + 0.623i)4-s + (1.90 + 1.17i)5-s − 3.01·6-s + (−2.21 − 2.21i)7-s + (−0.846 − 0.532i)8-s + (−4.76 − 3.79i)9-s + (−0.479 + 2.18i)10-s + (−3.58 + 4.49i)11-s + (−0.995 − 2.84i)12-s + (1.03 − 1.65i)13-s + (1.35 − 2.81i)14-s + (−5.23 + 4.24i)15-s + (0.222 − 0.974i)16-s + (−1.28 − 2.04i)17-s + ⋯ |
| L(s) = 1 | + (0.233 + 0.667i)2-s + (−0.575 + 1.64i)3-s + (−0.390 + 0.311i)4-s + (0.851 + 0.524i)5-s − 1.23·6-s + (−0.835 − 0.835i)7-s + (−0.299 − 0.188i)8-s + (−1.58 − 1.26i)9-s + (−0.151 + 0.690i)10-s + (−1.08 + 1.35i)11-s + (−0.287 − 0.821i)12-s + (0.287 − 0.458i)13-s + (0.362 − 0.752i)14-s + (−1.35 + 1.09i)15-s + (0.0556 − 0.243i)16-s + (−0.312 − 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.311256 - 0.853281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.311256 - 0.853281i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.330 - 0.943i)T \) |
| 5 | \( 1 + (-1.90 - 1.17i)T \) |
| 43 | \( 1 + (3.10 + 5.77i)T \) |
| good | 3 | \( 1 + (0.995 - 2.84i)T + (-2.34 - 1.87i)T^{2} \) |
| 7 | \( 1 + (2.21 + 2.21i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.58 - 4.49i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.03 + 1.65i)T + (-5.64 - 11.7i)T^{2} \) |
| 17 | \( 1 + (1.28 + 2.04i)T + (-7.37 + 15.3i)T^{2} \) |
| 19 | \( 1 + (-4.37 - 5.48i)T + (-4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (3.11 - 0.350i)T + (22.4 - 5.11i)T^{2} \) |
| 29 | \( 1 + (2.97 + 1.43i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.93 - 0.930i)T + (19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + (-8.41 - 8.41i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.16 - 1.04i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (5.92 + 0.668i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (11.0 - 6.94i)T + (22.9 - 47.7i)T^{2} \) |
| 59 | \( 1 + (-9.75 - 2.22i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-2.22 - 4.61i)T + (-38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (-3.93 - 0.443i)T + (65.3 + 14.9i)T^{2} \) |
| 71 | \( 1 + (-0.489 + 0.390i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-3.27 - 2.05i)T + (31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 - 17.4iT - 79T^{2} \) |
| 83 | \( 1 + (3.59 + 1.25i)T + (64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (4.10 - 1.97i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (0.799 + 7.09i)T + (-94.5 + 21.5i)T^{2} \) |
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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
−11.46994728068480154430425890184, −10.35218250197209253449906533231, −9.914290334998439284545193475782, −9.576584395407418522971800231632, −7.934070911517962571942044618644, −6.83788593702083104644542147393, −5.83890227356762730916119575510, −5.10114894877212646896863970770, −4.06766575135720687104856938688, −3.01240180971303110102548121312,
0.55318139344561633704635369621, 2.04783447032741304864699937921, 2.96162957145335114143552615642, 5.16213094182319796157355450966, 5.93182664801862227143035368817, 6.44242562515567933250316747933, 7.909469026681934517942109402180, 8.815798259239532387481014805419, 9.685288138933472256696915693535, 11.06387431214319942965188780543
