| L(s) = 1 | + (−8.02 − 4.63i)3-s + (−9.68 + 5.59i)5-s + (−5.50 − 48.6i)7-s + (2.44 + 4.22i)9-s + (−62.7 + 108. i)11-s − 130. i·13-s + 103.·15-s + (251. + 145. i)17-s + (−364. + 210. i)19-s + (−181. + 416. i)21-s + (417. + 723. i)23-s + (62.5 − 108. i)25-s + 705. i·27-s + 689.·29-s + (155. + 89.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.891 − 0.514i)3-s + (−0.387 + 0.223i)5-s + (−0.112 − 0.993i)7-s + (0.0301 + 0.0522i)9-s + (−0.518 + 0.898i)11-s − 0.770i·13-s + 0.460·15-s + (0.871 + 0.503i)17-s + (−1.01 + 0.583i)19-s + (−0.411 + 0.943i)21-s + (0.789 + 1.36i)23-s + (0.100 − 0.173i)25-s + 0.967i·27-s + 0.820·29-s + (0.161 + 0.0931i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.468785 + 0.351429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.468785 + 0.351429i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (9.68 - 5.59i)T \) |
| 7 | \( 1 + (5.50 + 48.6i)T \) |
| good | 3 | \( 1 + (8.02 + 4.63i)T + (40.5 + 70.1i)T^{2} \) |
| 11 | \( 1 + (62.7 - 108. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + 130. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (-251. - 145. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (364. - 210. i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-417. - 723. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 - 689.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-155. - 89.5i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-1.18e3 - 2.05e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + 8.95iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.41e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (-1.79e3 + 1.03e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (1.18e3 - 2.06e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.98e3 + 1.14e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.37e3 + 1.36e3i)T + (6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-945. + 1.63e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 8.10e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-831. - 480. i)T + (1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-4.18e3 - 7.25e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 7.33e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-1.69e3 + 975. i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 - 4.68e3iT - 8.85e7T^{2} \) |
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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
−12.60648788886325522053966704844, −11.72352122340749936315840543594, −10.64842134840740146468836070585, −9.950987610647614498794314695907, −8.108800529684535481950148117881, −7.23410312384142115351980653643, −6.22603207804117927606194135083, −4.91169901375682631391714915375, −3.39460788872667980018130615346, −1.18205696290835281035465503994,
0.30377053107423277089112898293, 2.70506451057036517852497020315, 4.52060498522021747755219085762, 5.47514287965341984237746562993, 6.52675282910896993787075299316, 8.202299542390947710662456371216, 9.078554729695972401171510174245, 10.42989849451324258951379750356, 11.28436939994337143148021203735, 12.03280336967888828585021940736
