| L(s) = 1 | + (−2.42 − 1.46i)2-s + (7.50 − 4.01i)3-s + (3.71 + 7.08i)4-s + (0.941 + 1.14i)5-s + (−24.0 − 1.27i)6-s + (2.81 − 1.88i)7-s + (1.37 − 22.5i)8-s + (25.2 − 37.7i)9-s + (−0.599 − 4.15i)10-s + (28.5 − 8.66i)11-s + (56.3 + 38.2i)12-s + (−2.68 − 2.20i)13-s + (−9.56 + 0.432i)14-s + (11.6 + 4.83i)15-s + (−36.3 + 52.6i)16-s + (62.9 − 26.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.855 − 0.517i)2-s + (1.44 − 0.771i)3-s + (0.464 + 0.885i)4-s + (0.0842 + 0.102i)5-s + (−1.63 − 0.0867i)6-s + (0.151 − 0.101i)7-s + (0.0607 − 0.998i)8-s + (0.934 − 1.39i)9-s + (−0.0189 − 0.131i)10-s + (0.782 − 0.237i)11-s + (1.35 + 0.920i)12-s + (−0.0573 − 0.0470i)13-s + (−0.182 + 0.00825i)14-s + (0.200 + 0.0831i)15-s + (−0.568 + 0.822i)16-s + (0.898 − 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.41621 - 1.19502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41621 - 1.19502i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.42 + 1.46i)T \) |
| good | 3 | \( 1 + (-7.50 + 4.01i)T + (15.0 - 22.4i)T^{2} \) |
| 5 | \( 1 + (-0.941 - 1.14i)T + (-24.3 + 122. i)T^{2} \) |
| 7 | \( 1 + (-2.81 + 1.88i)T + (131. - 316. i)T^{2} \) |
| 11 | \( 1 + (-28.5 + 8.66i)T + (1.10e3 - 739. i)T^{2} \) |
| 13 | \( 1 + (2.68 + 2.20i)T + (428. + 2.15e3i)T^{2} \) |
| 17 | \( 1 + (-62.9 + 26.0i)T + (3.47e3 - 3.47e3i)T^{2} \) |
| 19 | \( 1 + (-1.70 - 17.3i)T + (-6.72e3 + 1.33e3i)T^{2} \) |
| 23 | \( 1 + (83.5 - 16.6i)T + (1.12e4 - 4.65e3i)T^{2} \) |
| 29 | \( 1 + (-38.0 + 125. i)T + (-2.02e4 - 1.35e4i)T^{2} \) |
| 31 | \( 1 + (-29.8 - 29.8i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (301. + 29.6i)T + (4.96e4 + 9.88e3i)T^{2} \) |
| 41 | \( 1 + (-45.5 - 228. i)T + (-6.36e4 + 2.63e4i)T^{2} \) |
| 43 | \( 1 + (328. + 175. i)T + (4.41e4 + 6.61e4i)T^{2} \) |
| 47 | \( 1 + (-114. - 277. i)T + (-7.34e4 + 7.34e4i)T^{2} \) |
| 53 | \( 1 + (-147. - 484. i)T + (-1.23e5 + 8.27e4i)T^{2} \) |
| 59 | \( 1 + (-219. + 180. i)T + (4.00e4 - 2.01e5i)T^{2} \) |
| 61 | \( 1 + (-261. - 488. i)T + (-1.26e5 + 1.88e5i)T^{2} \) |
| 67 | \( 1 + (166. + 312. i)T + (-1.67e5 + 2.50e5i)T^{2} \) |
| 71 | \( 1 + (158. + 237. i)T + (-1.36e5 + 3.30e5i)T^{2} \) |
| 73 | \( 1 + (-767. - 513. i)T + (1.48e5 + 3.59e5i)T^{2} \) |
| 79 | \( 1 + (438. - 1.05e3i)T + (-3.48e5 - 3.48e5i)T^{2} \) |
| 83 | \( 1 + (-833. + 82.0i)T + (5.60e5 - 1.11e5i)T^{2} \) |
| 89 | \( 1 + (-819. - 163. i)T + (6.51e5 + 2.69e5i)T^{2} \) |
| 97 | \( 1 + (1.29e3 + 1.29e3i)T + 9.12e5iT^{2} \) |
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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
−12.50874975015314414997281477128, −11.77787858036060194724662074601, −10.25391616016619138858669402028, −9.325102550816599188418121646220, −8.369536631933761425734694407422, −7.65188248814715372783493878206, −6.53527937318868760678953992738, −3.79522361822244153077840366705, −2.60606401842715177502735148109, −1.26617864794557674374754198849,
1.84043819493280665407115236743, 3.55327102369216168044272060062, 5.17579060069284398422414386283, 6.89590820389966844574737907766, 8.107689277872858440915517348496, 8.842115996376247822631050631563, 9.690986839258311344564445133481, 10.45921467937348297369232424235, 11.90422216001183887157486385100, 13.55769592909778495108476766335
