| L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (−0.959 + 0.281i)6-s + (1.55 + 0.996i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.0871 + 0.605i)11-s + (0.142 + 0.989i)12-s + (4.39 − 2.82i)13-s + (1.20 − 1.39i)14-s + (0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (0.716 − 0.210i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.292 + 0.337i)5-s + (−0.391 + 0.115i)6-s + (0.586 + 0.376i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.266 − 0.170i)10-s + (0.0262 + 0.182i)11-s + (0.0410 + 0.285i)12-s + (1.21 − 0.783i)13-s + (0.322 − 0.372i)14-s + (0.107 − 0.234i)15-s + (0.210 + 0.135i)16-s + (0.173 − 0.0510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29770 - 0.980687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29770 - 0.980687i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-4.77 + 0.464i)T \) |
| good | 7 | \( 1 + (-1.55 - 0.996i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.0871 - 0.605i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-4.39 + 2.82i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.716 + 0.210i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.112 - 0.0330i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.21 + 0.356i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.68 + 8.06i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.14 - 1.31i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.17 - 3.66i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.41 + 3.09i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + (-2.36 - 1.52i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.82 - 1.81i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.89 + 4.15i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.108 - 0.753i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.0729 - 0.507i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 4.01i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (0.228 - 0.146i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (5.55 - 6.41i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.21 + 7.05i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.14 - 5.94i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.49315391208725652114776879416, −9.580562510056594331998018106152, −8.520545410364049291377817357169, −7.87086866412398105375299333830, −6.60376333491889620387982245643, −5.74713196814695793478902378368, −4.83895102908707697145408366352, −3.46420623587017350055689042692, −2.35683533176636944059460726137, −1.11345853206617258958862321074,
1.30430093229683529312094630102, 3.35986931729668413123396951639, 4.44235201122423496338293379005, 5.16415497731265431758013658639, 6.18488107989994562157916725240, 6.97139512855881586496046298364, 8.200170707759645358896711359232, 8.808337664830230195106457293251, 9.632512496326103319020439406351, 10.68830130857713322703907641075
