| L(s) = 1 | + (−2.38 − 1.21i)2-s + (−3.57 − 0.566i)3-s + (1.85 + 2.54i)4-s + (−4.45 − 2.27i)5-s + (7.83 + 5.69i)6-s + (6.54 − 6.54i)7-s + (0.355 + 2.24i)8-s + (3.91 + 1.27i)9-s + (7.83 + 10.8i)10-s + (−3.18 − 9.80i)11-s + (−5.17 − 10.1i)12-s + (−11.3 + 5.77i)13-s + (−23.5 + 7.65i)14-s + (14.6 + 10.6i)15-s + (5.77 − 17.7i)16-s + (0.578 − 0.0915i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 − 0.606i)2-s + (−1.19 − 0.188i)3-s + (0.462 + 0.636i)4-s + (−0.890 − 0.455i)5-s + (1.30 + 0.948i)6-s + (0.935 − 0.935i)7-s + (0.0444 + 0.280i)8-s + (0.434 + 0.141i)9-s + (0.783 + 1.08i)10-s + (−0.289 − 0.891i)11-s + (−0.431 − 0.846i)12-s + (−0.871 + 0.444i)13-s + (−1.68 + 0.546i)14-s + (0.975 + 0.711i)15-s + (0.360 − 1.11i)16-s + (0.0340 − 0.00538i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0471229 - 0.249871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0471229 - 0.249871i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.45 + 2.27i)T \) |
| good | 2 | \( 1 + (2.38 + 1.21i)T + (2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (3.57 + 0.566i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-6.54 + 6.54i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.18 + 9.80i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (11.3 - 5.77i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-0.578 + 0.0915i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-1.57 + 2.16i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-16.5 + 32.4i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-9.15 - 12.6i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (15.1 + 10.9i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (26.1 + 51.3i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-0.808 + 2.48i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (7.76 + 7.76i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (6.13 - 38.7i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-39.0 - 6.18i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-0.494 - 0.160i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-7.44 - 22.9i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-6.71 + 1.06i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-80.2 + 58.3i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-27.3 + 53.6i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-2.28 - 3.14i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (10.3 + 65.4i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (49.5 - 16.0i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-0.610 + 3.85i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−16.97955442545312657594765490447, −16.47090080243887852246291482111, −14.37389738562518082590537038238, −12.32760316387274247302662145855, −11.23785056132812567603126528119, −10.68625275369581608284251028995, −8.707564344676892490921017461846, −7.39178216663454693823579173031, −4.93602564596353106221573162120, −0.59588774193697738264585905627,
5.15639626532002690639409012138, 7.07494462927535818776735728409, 8.267769542691984107898920695577, 10.01690132627934471679503044316, 11.32490820185808803951179099993, 12.29607418552064484739334334592, 15.02082992755766604770966463697, 15.64447360631118295264181499855, 17.02701138745524775739740952990, 17.76881755723244691635609175996
