| L(s) = 1 | + (−2.23 + 0.726i)2-s + (−0.587 − 0.809i)3-s + (2.85 − 2.07i)4-s + (0.725 + 2.11i)5-s + (1.90 + 1.38i)6-s + 3.48i·7-s + (−2.10 + 2.90i)8-s + (−0.309 + 0.951i)9-s + (−3.15 − 4.20i)10-s + (0.905 + 2.78i)11-s + (−3.35 − 1.08i)12-s + (1.78 + 0.579i)13-s + (−2.52 − 7.78i)14-s + (1.28 − 1.83i)15-s + (0.427 − 1.31i)16-s + (3.98 − 5.48i)17-s + ⋯ |
| L(s) = 1 | + (−1.58 + 0.513i)2-s + (−0.339 − 0.467i)3-s + (1.42 − 1.03i)4-s + (0.324 + 0.945i)5-s + (0.776 + 0.564i)6-s + 1.31i·7-s + (−0.745 + 1.02i)8-s + (−0.103 + 0.317i)9-s + (−0.998 − 1.32i)10-s + (0.273 + 0.840i)11-s + (−0.968 − 0.314i)12-s + (0.494 + 0.160i)13-s + (−0.676 − 2.08i)14-s + (0.331 − 0.472i)15-s + (0.106 − 0.328i)16-s + (0.967 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.342607 + 0.274799i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.342607 + 0.274799i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 + (-0.725 - 2.11i)T \) |
| good | 2 | \( 1 + (2.23 - 0.726i)T + (1.61 - 1.17i)T^{2} \) |
| 7 | \( 1 - 3.48iT - 7T^{2} \) |
| 11 | \( 1 + (-0.905 - 2.78i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 0.579i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.98 + 5.48i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.38 + 1.73i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (5.22 - 1.69i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.06 + 1.50i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.338 + 0.245i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.98 - 1.61i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.518 + 1.59i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + (-4.40 - 6.06i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.18 - 3.00i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.19 + 6.76i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.98 + 6.12i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.90 + 8.12i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (0.589 - 0.428i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.41 - 1.11i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.48 + 1.80i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.94 - 8.18i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (0.0888 + 0.273i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.11 - 8.42i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−15.19676856260842242253515688522, −13.98033524331950922183552814072, −12.22194069645451106661417828898, −11.26819294153268337522090011328, −10.00881171317398258407465969781, −9.165364042221746333908681857257, −7.82999803109319075000280401261, −6.79437269818085932802266263525, −5.79046699521776151423315181399, −2.21998836829123061525635677545,
1.10927952017439266524446518883, 3.97119650213413439514429003381, 6.07868138148418199983747586329, 7.911303059110058415371347042140, 8.716406632367497739328146001097, 10.04603946738457520437370207939, 10.52555442854972098865642598714, 11.70005748169394532004574179123, 12.96356802749284088947035454841, 14.30981179259254991022623272169
