L(s) = 1 | + (−7.57 − 3.13i)3-s + (−8.03 − 19.4i)5-s + (1.85 − 1.85i)7-s + (28.3 + 28.3i)9-s + (−8.63 + 3.57i)11-s + (−11.5 + 27.9i)13-s + 172. i·15-s − 7.99i·17-s + (−5.76 + 13.9i)19-s + (−19.8 + 8.20i)21-s + (−60.2 − 60.2i)23-s + (−223. + 223. i)25-s + (−41.2 − 99.4i)27-s + (167. + 69.3i)29-s + 225.·31-s + ⋯ |
L(s) = 1 | + (−1.45 − 0.603i)3-s + (−0.718 − 1.73i)5-s + (0.0999 − 0.0999i)7-s + (1.05 + 1.05i)9-s + (−0.236 + 0.0979i)11-s + (−0.247 + 0.597i)13-s + 2.96i·15-s − 0.114i·17-s + (−0.0695 + 0.167i)19-s + (−0.205 + 0.0853i)21-s + (−0.546 − 0.546i)23-s + (−1.78 + 1.78i)25-s + (−0.293 − 0.709i)27-s + (1.07 + 0.444i)29-s + 1.30·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0421230 + 0.0574658i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0421230 + 0.0574658i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (7.57 + 3.13i)T + (19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (8.03 + 19.4i)T + (-88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (-1.85 + 1.85i)T - 343iT^{2} \) |
| 11 | \( 1 + (8.63 - 3.57i)T + (941. - 941. i)T^{2} \) |
| 13 | \( 1 + (11.5 - 27.9i)T + (-1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 + 7.99iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (5.76 - 13.9i)T + (-4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (60.2 + 60.2i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-167. - 69.3i)T + (1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-0.431 - 1.04i)T + (-3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (275. + 275. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (257. - 106. i)T + (5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 + 51.3iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (2.98 - 1.23i)T + (1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (101. + 244. i)T + (-1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (270. + 111. i)T + (1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (778. + 322. i)T + (2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (484. - 484. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (212. + 212. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 593. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (320. - 773. i)T + (-4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-435. + 435. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + 570.T + 9.12e5T^{2} \) |
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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
−12.12026430480763054698712613713, −11.58329132284101015509885882139, −10.24712692637165269839294328432, −8.788160060435697840656634165390, −7.75733761532722814106111942317, −6.46464671506500618437363224304, −5.16630055967086416875574008184, −4.44573526349238147850573117155, −1.30611018587938653550858511905, −0.04837821624346476269289734211,
3.07446109384852445646649008735, 4.52130880434492146039730264164, 5.93952527339901391327614437338, 6.80443025531803635271355293177, 8.011663990340480324039428070290, 10.18057574479742561278366181184, 10.41299525362709484801309044274, 11.59963222331296672536130769252, 11.90715430684543874254685960903, 13.62047229785038842121974785663