| L(s) = 1 | + (−0.830 − 1.81i)2-s + (−2.87 + 0.845i)3-s + (−2.61 + 3.02i)4-s + (−3.38 − 2.17i)5-s + (3.92 + 4.53i)6-s + (−3.30 − 22.9i)7-s + (7.67 + 2.25i)8-s + (7.57 − 4.86i)9-s + (−1.14 + 7.97i)10-s + (−8.80 + 19.2i)11-s + (4.98 − 10.9i)12-s + (−9.66 + 67.2i)13-s + (−39.0 + 25.1i)14-s + (11.5 + 3.40i)15-s + (−2.27 − 15.8i)16-s + (45.5 + 52.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.303 − 0.194i)5-s + (0.267 + 0.308i)6-s + (−0.178 − 1.24i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0362 + 0.252i)10-s + (−0.241 + 0.528i)11-s + (0.119 − 0.262i)12-s + (−0.206 + 1.43i)13-s + (−0.746 + 0.479i)14-s + (0.199 + 0.0585i)15-s + (−0.0355 − 0.247i)16-s + (0.650 + 0.750i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.423317 + 0.312526i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.423317 + 0.312526i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.830 + 1.81i)T \) |
| 3 | \( 1 + (2.87 - 0.845i)T \) |
| 23 | \( 1 + (-109. - 14.4i)T \) |
| good | 5 | \( 1 + (3.38 + 2.17i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (3.30 + 22.9i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (8.80 - 19.2i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (9.66 - 67.2i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (-45.5 - 52.5i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (100. - 116. i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-140. - 162. i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (48.1 + 14.1i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (288. - 185. i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (130. + 83.6i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-327. + 96.2i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 + 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-69.7 - 485. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (23.9 - 166. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (383. + 112. i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (289. + 633. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (141. + 309. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (717. - 828. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-94.5 + 657. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-133. + 85.7i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-1.34e3 + 394. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-664. - 427. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−12.55023754681877115865626384249, −11.98013137815248812628961501731, −10.58588286549604462051548740173, −10.28226823970129951145666551893, −8.907173652033094868855643754839, −7.61130442696946828662299250127, −6.51117258694704879759093675342, −4.65035435994678126351925739824, −3.78035422208104624410982121647, −1.50436000178862461273507171782,
0.32341706230834453603489950561, 2.84874384058370073459165369713, 5.04087275099669655441096751588, 5.85418533616949714346406139649, 7.06141865847005567106625065976, 8.214799529981111260996082526226, 9.193332962508738699693308481758, 10.47980859437325547873675094581, 11.44913658714719285392525587034, 12.56214055332862171817408875127
