| L(s) = 1 | + (0.215 − 2.46i)2-s + (0.236 + 0.507i)3-s + (−4.05 − 0.714i)4-s + (1.40 − 1.73i)5-s + (1.30 − 0.473i)6-s + (−2.99 + 0.801i)7-s + (−1.35 + 5.05i)8-s + (1.72 − 2.05i)9-s + (−3.97 − 3.84i)10-s + (2.71 + 4.70i)11-s + (−0.596 − 2.22i)12-s + (1.77 + 0.828i)13-s + (1.32 + 7.53i)14-s + (1.21 + 0.304i)15-s + (4.41 + 1.60i)16-s + (−1.67 − 0.146i)17-s + ⋯ |
| L(s) = 1 | + (0.152 − 1.74i)2-s + (0.136 + 0.292i)3-s + (−2.02 − 0.357i)4-s + (0.630 − 0.776i)5-s + (0.531 − 0.193i)6-s + (−1.13 + 0.302i)7-s + (−0.478 + 1.78i)8-s + (0.575 − 0.686i)9-s + (−1.25 − 1.21i)10-s + (0.818 + 1.41i)11-s + (−0.172 − 0.642i)12-s + (0.492 + 0.229i)13-s + (0.355 + 2.01i)14-s + (0.313 + 0.0786i)15-s + (1.10 + 0.402i)16-s + (−0.405 − 0.0354i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.472885 - 0.943411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472885 - 0.943411i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.40 + 1.73i)T \) |
| 19 | \( 1 + (-2.83 - 3.30i)T \) |
| good | 2 | \( 1 + (-0.215 + 2.46i)T + (-1.96 - 0.347i)T^{2} \) |
| 3 | \( 1 + (-0.236 - 0.507i)T + (-1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (2.99 - 0.801i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.71 - 4.70i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.77 - 0.828i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (1.67 + 0.146i)T + (16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (1.35 + 1.93i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.236 - 0.198i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.68 + 2.12i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.08 - 3.08i)T - 37iT^{2} \) |
| 41 | \( 1 + (3.63 - 9.98i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.69 + 3.28i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (0.255 + 2.92i)T + (-46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (5.67 - 3.97i)T + (18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-1.92 + 1.61i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.00 + 5.67i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.26 - 0.197i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-8.76 + 1.54i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-11.1 + 5.20i)T + (46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (14.4 + 5.27i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.982 + 3.66i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (3.79 - 1.38i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.18 + 13.4i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−13.07853389843389875321563808436, −12.52656717915652217335996693330, −11.78405418882004846667277989314, −10.04798419776281758351566159555, −9.686139315636248952308637068783, −8.922886249917338751098559603093, −6.47637006979439669363154520358, −4.66226173209518504206889961944, −3.54322273553511967661825383173, −1.69481109202971901666727797731,
3.55835646468649110358242973499, 5.54719167645593907884477290851, 6.54954761319409496577290059457, 7.18392725483830921235383710942, 8.591569533105151172680009593632, 9.654336002382021498441296605382, 11.00049133750747546263823128281, 13.04319823421820027475766555886, 13.68868204190381455989423328102, 14.19583258081686457077001310859
