| L(s) = 1 | + (−1.20 + 1.00i)2-s + (−2.14 − 0.781i)3-s + (0.0809 − 0.459i)4-s + (−0.173 − 0.984i)5-s + (3.37 − 1.22i)6-s + (2.00 − 3.47i)7-s + (−1.20 − 2.08i)8-s + (1.70 + 1.42i)9-s + (1.20 + 1.00i)10-s + (−1.38 − 2.39i)11-s + (−0.532 + 0.922i)12-s + (−2.61 + 0.953i)13-s + (1.09 + 6.20i)14-s + (−0.396 + 2.25i)15-s + (4.43 + 1.61i)16-s + (−2.76 + 2.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.850 + 0.713i)2-s + (−1.23 − 0.451i)3-s + (0.0404 − 0.229i)4-s + (−0.0776 − 0.440i)5-s + (1.37 − 0.501i)6-s + (0.758 − 1.31i)7-s + (−0.425 − 0.737i)8-s + (0.567 + 0.475i)9-s + (0.380 + 0.319i)10-s + (−0.417 − 0.722i)11-s + (−0.153 + 0.266i)12-s + (−0.726 + 0.264i)13-s + (0.292 + 1.65i)14-s + (−0.102 + 0.581i)15-s + (1.10 + 0.403i)16-s + (−0.670 + 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.268185 - 0.206258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.268185 - 0.206258i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.79 + 3.97i)T \) |
| good | 2 | \( 1 + (1.20 - 1.00i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (2.14 + 0.781i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-2.00 + 3.47i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.38 + 2.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.61 - 0.953i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.76 - 2.31i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.237 + 1.34i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (7.28 + 6.11i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.776 - 1.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 + (-6.21 - 2.26i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.08 + 6.15i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.97 - 5.85i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.684 - 3.88i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.76 + 2.32i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.30 - 7.37i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (11.1 + 9.36i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.576 + 3.26i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-9.40 - 3.42i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.82 - 0.666i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.809 - 1.40i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 4.19i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (8.87 - 7.45i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.61987560990400424294927728721, −12.68435927308867512031298679082, −11.44792513709340196291625453770, −10.67849587914798551528529728449, −9.232170818151687425202906799534, −7.87129376293862254748368101756, −7.14873285021946491108570464131, −5.92119966635454052775169975658, −4.41433968694849753886990624819, −0.61015637246196384439446783475,
2.31119102367600441797733906255, 5.01149713115591120810990620522, 5.79078227095199108834625104567, 7.68945528899398886971914451797, 9.147399283186567065200921012893, 10.07522776012345031809091295713, 11.08597276844240410538579949031, 11.64579713723763758379317719035, 12.51162866720314243245283772080, 14.57828854956164941597190684194
