| L(s) = 1 | + (0.284 + 1.97i)2-s + (2.60 − 5.70i)3-s + (−3.83 + 1.12i)4-s + (6.82 − 7.87i)5-s + (12.0 + 3.53i)6-s + (9.96 − 6.40i)7-s + (−3.32 − 7.27i)8-s + (−8.03 − 9.27i)9-s + (17.5 + 11.2i)10-s + (−5.03 + 35.0i)11-s + (−3.56 + 24.8i)12-s + (3.24 + 2.08i)13-s + (15.5 + 17.9i)14-s + (−27.1 − 59.4i)15-s + (13.4 − 8.65i)16-s + (29.6 + 8.71i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (0.500 − 1.09i)3-s + (−0.479 + 0.140i)4-s + (0.610 − 0.704i)5-s + (0.818 + 0.240i)6-s + (0.538 − 0.345i)7-s + (−0.146 − 0.321i)8-s + (−0.297 − 0.343i)9-s + (0.554 + 0.356i)10-s + (−0.138 + 0.960i)11-s + (−0.0858 + 0.596i)12-s + (0.0692 + 0.0445i)13-s + (0.296 + 0.341i)14-s + (−0.467 − 1.02i)15-s + (0.210 − 0.135i)16-s + (0.423 + 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.66605 - 0.243391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66605 - 0.243391i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.284 - 1.97i)T \) |
| 23 | \( 1 + (94.6 + 56.6i)T \) |
| good | 3 | \( 1 + (-2.60 + 5.70i)T + (-17.6 - 20.4i)T^{2} \) |
| 5 | \( 1 + (-6.82 + 7.87i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-9.96 + 6.40i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (5.03 - 35.0i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (-3.24 - 2.08i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-29.6 - 8.71i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (128. - 37.8i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (60.1 + 17.6i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-107. - 235. i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-214. - 247. i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-45.2 + 52.2i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-123. + 270. i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 + 502.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-485. + 312. i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (376. + 241. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-147. - 322. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (143. + 996. i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (107. + 746. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (679. - 199. i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-96.0 - 61.7i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (169. + 195. i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-141. + 310. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-412. + 475. i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
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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
−14.96478793068914567574339309562, −14.02660784335449433139114524492, −13.05683216416426201127259229018, −12.32417684136152639890938770923, −10.18187418689837887636547592598, −8.622845501116646170352954508157, −7.70477340491876142409212719434, −6.40412294498257651498905428262, −4.68876261243636678587369932277, −1.73493959810407376890466520748,
2.61872221405972442469632727576, 4.17981582738332652792015705097, 5.92389668639890767805130505572, 8.339858026494955565115755612488, 9.542250549494396555982287047727, 10.50881181426899418390855699903, 11.44355087171978734118198261754, 13.16713774857629556935596229243, 14.36767818907163379711807749233, 14.98581451593820101791059289516
