| L(s) = 1 | + (−0.806 + 1.16i)2-s + (1.60 + 0.559i)3-s + (−0.699 − 1.87i)4-s + (1.69 + 0.386i)5-s + (−1.94 + 1.40i)6-s + (0.260 + 0.541i)7-s + (2.74 + 0.697i)8-s + (−0.0983 − 0.0784i)9-s + (−1.81 + 1.65i)10-s + (0.116 − 1.03i)11-s + (−0.0706 − 3.38i)12-s + (−3.87 + 3.08i)13-s + (−0.838 − 0.133i)14-s + (2.49 + 1.56i)15-s + (−3.02 + 2.62i)16-s + (−1.06 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.570 + 0.821i)2-s + (0.923 + 0.323i)3-s + (−0.349 − 0.936i)4-s + (0.756 + 0.172i)5-s + (−0.792 + 0.574i)6-s + (0.0984 + 0.204i)7-s + (0.969 + 0.246i)8-s + (−0.0327 − 0.0261i)9-s + (−0.573 + 0.523i)10-s + (0.0350 − 0.310i)11-s + (−0.0204 − 0.978i)12-s + (−1.07 + 0.856i)13-s + (−0.224 − 0.0356i)14-s + (0.643 + 0.404i)15-s + (−0.755 + 0.655i)16-s + (−0.257 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.910441 + 0.569562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.910441 + 0.569562i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.806 - 1.16i)T \) |
| 29 | \( 1 + (-1.22 + 5.24i)T \) |
| good | 3 | \( 1 + (-1.60 - 0.559i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-1.69 - 0.386i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (-0.260 - 0.541i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-0.116 + 1.03i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (3.87 - 3.08i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (1.06 - 1.06i)T - 17iT^{2} \) |
| 19 | \( 1 + (1.36 + 3.91i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (-7.10 + 1.62i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (3.05 - 1.91i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (-5.84 + 0.658i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (3.60 + 3.60i)T + 41iT^{2} \) |
| 43 | \( 1 + (4.74 - 7.54i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (0.738 + 0.0832i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (2.05 - 9.00i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 - 1.92iT - 59T^{2} \) |
| 61 | \( 1 + (4.10 - 11.7i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (-0.940 + 1.17i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-7.06 - 8.85i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-2.10 + 3.34i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (16.2 - 1.83i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (-2.80 + 5.81i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.413 - 0.657i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (4.94 + 14.1i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−14.07347080194299614401505607882, −13.20871937832036635868574186232, −11.42499023773419619150050749222, −10.09622136091959644025998262904, −9.246863935150558348035766492375, −8.612841474074656879714250635611, −7.22712750796698235456613165846, −6.07883188944560705402902065916, −4.61285465488537990584439292541, −2.42949894046672591350540802233,
1.93266144311174935982517536595, 3.19694077444732594739946031394, 5.10204360153532304580700664435, 7.22156499726174861623762061972, 8.159038221938119877127518589155, 9.231524320947591983984467846450, 10.01987459258572852388885525887, 11.14959895856929946936180588093, 12.55312233070718072314805886445, 13.18660534148785204498481362304
