| L(s) = 1 | + (1.75 + 1.47i)2-s + (−3.10 + 1.13i)3-s + (0.563 + 3.19i)4-s + (0.173 − 0.984i)5-s + (−7.11 − 2.59i)6-s + (1.46 + 2.54i)7-s + (−1.42 + 2.46i)8-s + (6.08 − 5.10i)9-s + (1.75 − 1.47i)10-s + (0.288 − 0.500i)11-s + (−5.36 − 9.29i)12-s + (0.629 + 0.229i)13-s + (−1.16 + 6.62i)14-s + (0.574 + 3.25i)15-s + (−0.0331 + 0.0120i)16-s + (−0.269 − 0.226i)17-s + ⋯ |
| L(s) = 1 | + (1.24 + 1.04i)2-s + (−1.79 + 0.653i)3-s + (0.281 + 1.59i)4-s + (0.0776 − 0.440i)5-s + (−2.90 − 1.05i)6-s + (0.555 + 0.962i)7-s + (−0.503 + 0.872i)8-s + (2.02 − 1.70i)9-s + (0.554 − 0.465i)10-s + (0.0870 − 0.150i)11-s + (−1.54 − 2.68i)12-s + (0.174 + 0.0635i)13-s + (−0.312 + 1.77i)14-s + (0.148 + 0.840i)15-s + (−0.00829 + 0.00302i)16-s + (−0.0653 − 0.0548i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.647388 + 0.982783i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647388 + 0.982783i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (3.86 + 2.02i)T \) |
| good | 2 | \( 1 + (-1.75 - 1.47i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (3.10 - 1.13i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.46 - 2.54i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.288 + 0.500i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.629 - 0.229i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.269 + 0.226i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.715 + 4.05i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.30 + 1.93i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.148 + 0.257i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.30T + 37T^{2} \) |
| 41 | \( 1 + (2.51 - 0.913i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.14 - 6.49i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.45 + 7.09i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.713 + 4.04i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.467 - 0.392i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.178 - 1.01i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.55 - 2.14i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.29 - 13.0i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (6.70 - 2.44i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (1.44 - 0.527i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (6.65 + 11.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.17 - 2.24i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.34 - 7.84i)T + (16.8 + 95.5i)T^{2} \) |
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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
−14.65659620860875013288580135227, −13.14110913444294067537951343446, −12.24198071189871269310342577668, −11.59537355743867894088991983400, −10.31103415241885663580180750608, −8.653721213591577271292876854990, −6.81045412329225065949363743836, −5.89527381597697796353753250618, −5.10380850119003676340447693279, −4.24915594918939947076845729427,
1.59961005172365643932825124326, 4.09854972517032123497977337494, 5.21296993796896472378315405310, 6.31130994598573065383531080259, 7.48592988238488705478228121537, 10.48727763718729936657427778603, 10.66662062743066149645532318418, 11.71413682252650503965971682708, 12.39914679427681230460296881865, 13.39025072455994538712172657838
