| L(s) = 1 | + (1.42 + 0.417i)2-s + (4.44 − 5.13i)3-s + (−4.88 − 3.13i)4-s + (−1.63 + 11.3i)5-s + (8.47 − 5.44i)6-s + (8.78 + 19.2i)7-s + (−13.4 − 15.4i)8-s + (−2.72 − 18.9i)9-s + (−7.07 + 15.4i)10-s + (−40.8 + 11.9i)11-s + (−37.8 + 11.1i)12-s + (26.8 − 58.7i)13-s + (4.46 + 31.0i)14-s + (51.1 + 58.9i)15-s + (6.67 + 14.6i)16-s + (34.3 − 22.1i)17-s + ⋯ |
| L(s) = 1 | + (0.502 + 0.147i)2-s + (0.856 − 0.987i)3-s + (−0.610 − 0.392i)4-s + (−0.146 + 1.01i)5-s + (0.576 − 0.370i)6-s + (0.474 + 1.03i)7-s + (−0.592 − 0.683i)8-s + (−0.100 − 0.701i)9-s + (−0.223 + 0.489i)10-s + (−1.11 + 0.328i)11-s + (−0.909 + 0.267i)12-s + (0.572 − 1.25i)13-s + (0.0851 + 0.592i)14-s + (0.879 + 1.01i)15-s + (0.104 + 0.228i)16-s + (0.490 − 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.47985 - 0.242206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47985 - 0.242206i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-97.3 - 51.9i)T \) |
| good | 2 | \( 1 + (-1.42 - 0.417i)T + (6.73 + 4.32i)T^{2} \) |
| 3 | \( 1 + (-4.44 + 5.13i)T + (-3.84 - 26.7i)T^{2} \) |
| 5 | \( 1 + (1.63 - 11.3i)T + (-119. - 35.2i)T^{2} \) |
| 7 | \( 1 + (-8.78 - 19.2i)T + (-224. + 259. i)T^{2} \) |
| 11 | \( 1 + (40.8 - 11.9i)T + (1.11e3 - 719. i)T^{2} \) |
| 13 | \( 1 + (-26.8 + 58.7i)T + (-1.43e3 - 1.66e3i)T^{2} \) |
| 17 | \( 1 + (-34.3 + 22.1i)T + (2.04e3 - 4.46e3i)T^{2} \) |
| 19 | \( 1 + (85.8 + 55.1i)T + (2.84e3 + 6.23e3i)T^{2} \) |
| 29 | \( 1 + (75.5 - 48.5i)T + (1.01e4 - 2.21e4i)T^{2} \) |
| 31 | \( 1 + (152. + 175. i)T + (-4.23e3 + 2.94e4i)T^{2} \) |
| 37 | \( 1 + (12.9 + 90.0i)T + (-4.86e4 + 1.42e4i)T^{2} \) |
| 41 | \( 1 + (12.0 - 83.6i)T + (-6.61e4 - 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-29.3 + 33.8i)T + (-1.13e4 - 7.86e4i)T^{2} \) |
| 47 | \( 1 - 326.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (4.62 + 10.1i)T + (-9.74e4 + 1.12e5i)T^{2} \) |
| 59 | \( 1 + (319. - 699. i)T + (-1.34e5 - 1.55e5i)T^{2} \) |
| 61 | \( 1 + (-505. - 583. i)T + (-3.23e4 + 2.24e5i)T^{2} \) |
| 67 | \( 1 + (1.03e3 + 304. i)T + (2.53e5 + 1.62e5i)T^{2} \) |
| 71 | \( 1 + (217. + 63.8i)T + (3.01e5 + 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-587. - 377. i)T + (1.61e5 + 3.53e5i)T^{2} \) |
| 79 | \( 1 + (-147. + 323. i)T + (-3.22e5 - 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-40.0 - 278. i)T + (-5.48e5 + 1.61e5i)T^{2} \) |
| 89 | \( 1 + (168. - 194. i)T + (-1.00e5 - 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-150. + 1.04e3i)T + (-8.75e5 - 2.57e5i)T^{2} \) |
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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
−18.10667491730730516727536990944, −15.18564229710359019558291540116, −14.90544369568535139466997128584, −13.46015113117647535826710639512, −12.70510001327635450437284581707, −10.70879502217928760770298868799, −8.791907614024264141217676376362, −7.44013465786981393755650267323, −5.57072289457261409698501131323, −2.79413047991421956770977430478,
3.80780379059166515046811419319, 4.80933808725663540902112620426, 8.182892241154164331356625412301, 9.053039093381673283805256646831, 10.70359861096109268632745430794, 12.61048652733823225436669518004, 13.73459864369453371727846232227, 14.65698068895870311429912517450, 16.22509376453060378661637860170, 17.06193846038053000274688369509
