| L(s) = 1 | + (−1.20 + 0.212i)2-s + (−0.517 − 0.616i)3-s + (−0.477 + 0.173i)4-s + (2.13 + 0.670i)5-s + (0.752 + 0.631i)6-s + (3.28 + 1.89i)7-s + (2.65 − 1.53i)8-s + (0.408 − 2.31i)9-s + (−2.70 − 0.354i)10-s + (0.618 + 1.07i)11-s + (0.353 + 0.204i)12-s + (−2.22 + 2.64i)13-s + (−4.35 − 1.58i)14-s + (−0.689 − 1.66i)15-s + (−2.08 + 1.75i)16-s + (2.96 − 0.522i)17-s + ⋯ |
| L(s) = 1 | + (−0.850 + 0.149i)2-s + (−0.298 − 0.355i)3-s + (−0.238 + 0.0868i)4-s + (0.953 + 0.300i)5-s + (0.307 + 0.257i)6-s + (1.24 + 0.716i)7-s + (0.937 − 0.541i)8-s + (0.136 − 0.772i)9-s + (−0.856 − 0.112i)10-s + (0.186 + 0.323i)11-s + (0.102 + 0.0589i)12-s + (−0.616 + 0.734i)13-s + (−1.16 − 0.423i)14-s + (−0.178 − 0.428i)15-s + (−0.522 + 0.438i)16-s + (0.718 − 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.695504 + 0.0849772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.695504 + 0.0849772i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.13 - 0.670i)T \) |
| 19 | \( 1 + (4.28 - 0.793i)T \) |
| good | 2 | \( 1 + (1.20 - 0.212i)T + (1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.517 + 0.616i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-3.28 - 1.89i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.618 - 1.07i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.22 - 2.64i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.96 + 0.522i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (2.10 + 5.77i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.744 + 4.22i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.55 - 4.41i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.13iT - 37T^{2} \) |
| 41 | \( 1 + (-4.08 + 3.42i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.12 - 8.57i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (7.19 + 1.26i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.13 - 3.13i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.141 - 0.804i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.01 - 2.18i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.995 - 0.175i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (12.8 + 4.67i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (7.11 + 8.47i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.06 + 0.889i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.18 - 1.26i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.06 + 1.73i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.01 + 0.531i)T + (91.1 - 33.1i)T^{2} \) |
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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
−14.31200172914800218381836684018, −12.83163252966248129930511517061, −11.94948406886653328562505807176, −10.59227725184206317772805882281, −9.505915320925671888913474154463, −8.678782885930509010080313006654, −7.38491533309747357899039610364, −6.13559846527721992984997789901, −4.60330175444546849359499246898, −1.83674972691443388723304891547,
1.62502339354937982886609193326, 4.62782240567786865294300562157, 5.48481732568462304671068199830, 7.61494202858623578489097212202, 8.480279535593034556689856222240, 9.861120579274296503312874896392, 10.44098268107228224536972053159, 11.36027994114025353439950633291, 13.12024569348545824505007627386, 13.92415641453018036462924936324
