| L(s) = 1 | + (0.236 − 1.39i)2-s + (1.15 + 1.83i)3-s + (−1.88 − 0.659i)4-s + (2.81 + 2.24i)5-s + (2.83 − 1.17i)6-s + (−2.89 − 0.661i)7-s + (−1.36 + 2.47i)8-s + (−0.738 + 1.53i)9-s + (3.79 − 3.38i)10-s + (0.943 − 2.69i)11-s + (−0.968 − 4.22i)12-s + (−1.89 − 3.93i)13-s + (−1.60 + 3.88i)14-s + (−0.872 + 7.74i)15-s + (3.13 + 2.48i)16-s + (−3.41 − 3.41i)17-s + ⋯ |
| L(s) = 1 | + (0.167 − 0.985i)2-s + (0.666 + 1.06i)3-s + (−0.944 − 0.329i)4-s + (1.25 + 1.00i)5-s + (1.15 − 0.479i)6-s + (−1.09 − 0.249i)7-s + (−0.482 + 0.875i)8-s + (−0.246 + 0.511i)9-s + (1.19 − 1.07i)10-s + (0.284 − 0.812i)11-s + (−0.279 − 1.22i)12-s + (−0.524 − 1.09i)13-s + (−0.429 + 1.03i)14-s + (−0.225 + 2.00i)15-s + (0.782 + 0.622i)16-s + (−0.827 − 0.827i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31306 - 0.143540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31306 - 0.143540i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.236 + 1.39i)T \) |
| 29 | \( 1 + (1.07 - 5.27i)T \) |
| good | 3 | \( 1 + (-1.15 - 1.83i)T + (-1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (-2.81 - 2.24i)T + (1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (2.89 + 0.661i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (-0.943 + 2.69i)T + (-8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (1.89 + 3.93i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (3.41 + 3.41i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.256 - 0.161i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (0.658 - 0.525i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-0.911 - 8.08i)T + (-30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (1.23 - 0.432i)T + (28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (-1.66 + 1.66i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.78 - 0.314i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (9.56 + 3.34i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (-3.16 + 3.97i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + 0.515iT - 59T^{2} \) |
| 61 | \( 1 + (10.6 - 6.71i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (-8.68 - 4.18i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (-2.66 + 1.28i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-12.4 - 1.40i)T + (71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (-4.20 + 1.47i)T + (61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (10.7 - 2.45i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.739 + 0.0832i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (3.88 + 2.44i)T + (42.0 + 87.3i)T^{2} \) |
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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
−13.67534856699054948173606173297, −12.66622408955293446467089371831, −10.99367893913358514829440978865, −10.22362687663881611405339922762, −9.678526269036618132953486088615, −8.813820906169183671773230593498, −6.62527477695165502550742389655, −5.24941415156877376472428822578, −3.43885010434372831842105927411, −2.79333003638805823204133507501,
2.09503261548384018337385250223, 4.48775938865255522849206056770, 6.08531761939323390603510561217, 6.77353278690690864915644590977, 8.097619652535209774772322897929, 9.296897712012497367865798363461, 9.627079416018120964996677979474, 12.31744798786661034366420926865, 12.94025388081647969527229626245, 13.49432313669503616268007341931
