| 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
−16.19224884137059967489144551364, −14.59543681727923282976358293873, −13.39830606476109052890475491733, −11.68442978032125672537903805163, −10.36485291734542553494756887831, −9.625289146492597392118716360618, −8.056086040450478827011421098576, −6.98910938834967324177768184257, −6.07133961041192912468456433790, −1.51082603664156141670277864912,
2.41756159418740217473334044149, 5.48303141057908662627759397655, 7.47649351432565134135919479480, 8.992400315194067536834872930907, 9.743624919292009080962151744966, 10.59338460892450861381649136476, 11.72805527166984780569820907659, 13.07008081590860911901993702709, 15.30439873856672007936541792124, 15.99799761197440130581242225561
