| L(s) = 1 | + (2.64 − 4.47i)3-s − 7.63·5-s + (−0.685 − 18.5i)7-s + (−12.9 − 23.6i)9-s + 33.1i·11-s + (−31.2 − 18.0i)13-s + (−20.2 + 34.1i)15-s + (−24.8 + 43.0i)17-s + (−16.5 + 9.54i)19-s + (−84.5 − 45.9i)21-s − 41.0i·23-s − 66.7·25-s + (−140. − 4.74i)27-s + (70.5 − 40.7i)29-s + (−185. + 106. i)31-s + ⋯ |
| L(s) = 1 | + (0.509 − 0.860i)3-s − 0.682·5-s + (−0.0370 − 0.999i)7-s + (−0.480 − 0.877i)9-s + 0.908i·11-s + (−0.667 − 0.385i)13-s + (−0.347 + 0.587i)15-s + (−0.354 + 0.614i)17-s + (−0.199 + 0.115i)19-s + (−0.878 − 0.477i)21-s − 0.372i·23-s − 0.534·25-s + (−0.999 − 0.0338i)27-s + (0.452 − 0.260i)29-s + (−1.07 + 0.619i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6716364985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6716364985\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.64 + 4.47i)T \) |
| 7 | \( 1 + (0.685 + 18.5i)T \) |
| good | 5 | \( 1 + 7.63T + 125T^{2} \) |
| 11 | \( 1 - 33.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (31.2 + 18.0i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (24.8 - 43.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (16.5 - 9.54i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 41.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-70.5 + 40.7i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (185. - 106. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (94.2 + 163. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-101. + 176. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-116. - 202. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (145. - 252. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (3.42 + 1.97i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (278. + 482. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-160. - 92.6i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (261. + 452. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 876. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (765. + 441. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-408. + 706. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-276. - 478. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-225. - 390. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-337. + 194. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.15294268377547884777329358364, −10.15419357478028829199877930175, −9.020491137337013193939231057208, −7.78239204644886622986465185345, −7.39956637355685639760587456684, −6.31122657888679822226932433992, −4.56903024694839084663087245875, −3.45837410187254999898039518523, −1.89247031972215829541875785587, −0.23515204209962637523416405718,
2.42775221633746262429080701483, 3.56084000954036750338281814918, 4.76918051891605685329185482083, 5.82419411250483118412877671492, 7.38021837576925155984345150975, 8.475040879538254989846898780457, 9.097702030153698138271067051351, 10.08075566494910927254223385025, 11.32895752801232082923125705744, 11.77217803451097188927336501021
