| L(s) = 1 | + (−2.50 − 0.772i)2-s + (−0.539 + 1.37i)3-s + (4.02 + 2.74i)4-s + (2.04 − 0.307i)5-s + (2.41 − 3.02i)6-s + (0.189 + 2.63i)7-s + (−4.69 − 5.89i)8-s + (0.598 + 0.555i)9-s + (−5.35 − 0.806i)10-s + (−1.36 + 1.27i)11-s + (−5.95 + 4.05i)12-s + (−0.677 − 2.96i)13-s + (1.56 − 6.75i)14-s + (−0.679 + 2.97i)15-s + (3.65 + 9.32i)16-s + (−0.104 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 − 0.546i)2-s + (−0.311 + 0.794i)3-s + (2.01 + 1.37i)4-s + (0.913 − 0.137i)5-s + (0.986 − 1.23i)6-s + (0.0717 + 0.997i)7-s + (−1.66 − 2.08i)8-s + (0.199 + 0.185i)9-s + (−1.69 − 0.255i)10-s + (−0.413 + 0.383i)11-s + (−1.71 + 1.17i)12-s + (−0.187 − 0.822i)13-s + (0.417 − 1.80i)14-s + (−0.175 + 0.768i)15-s + (0.914 + 2.33i)16-s + (−0.0254 − 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.408590 + 0.101187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.408590 + 0.101187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.189 - 2.63i)T \) |
| good | 2 | \( 1 + (2.50 + 0.772i)T + (1.65 + 1.12i)T^{2} \) |
| 3 | \( 1 + (0.539 - 1.37i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (-2.04 + 0.307i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (1.36 - 1.27i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.677 + 2.96i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.104 + 1.40i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-2.73 + 4.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0152 - 0.203i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-1.62 - 0.781i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (2.57 + 4.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.23 + 4.25i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (5.90 + 7.40i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-1.59 + 2.00i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-1.58 - 0.490i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (-5.98 - 4.07i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-8.25 - 1.24i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (5.45 - 3.72i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (1.78 + 3.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.02 - 2.89i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.168 - 0.0521i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (7.82 - 13.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.14 + 9.39i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.911 - 0.845i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−15.99799761197440130581242225561, −15.30439873856672007936541792124, −13.07008081590860911901993702709, −11.72805527166984780569820907659, −10.59338460892450861381649136476, −9.743624919292009080962151744966, −8.992400315194067536834872930907, −7.47649351432565134135919479480, −5.48303141057908662627759397655, −2.41756159418740217473334044149,
1.51082603664156141670277864912, 6.07133961041192912468456433790, 6.98910938834967324177768184257, 8.056086040450478827011421098576, 9.625289146492597392118716360618, 10.36485291734542553494756887831, 11.68442978032125672537903805163, 13.39830606476109052890475491733, 14.59543681727923282976358293873, 16.19224884137059967489144551364
