| L(s) = 1 | + (0.573 − 0.819i)2-s + (−0.342 − 0.939i)4-s + (−3.55 + 0.310i)5-s + (3.11 + 2.61i)7-s + (−0.965 − 0.258i)8-s + (−1.78 + 3.08i)10-s + (0.00168 + 0.00292i)11-s + (1.48 + 0.692i)13-s + (3.92 − 1.05i)14-s + (−0.766 + 0.642i)16-s + (6.72 − 3.13i)17-s + (5.40 − 3.78i)19-s + (1.50 + 3.23i)20-s + (0.00335 + 0.000293i)22-s + (1.87 + 7.01i)23-s + ⋯ |
| L(s) = 1 | + (0.405 − 0.579i)2-s + (−0.171 − 0.469i)4-s + (−1.58 + 0.138i)5-s + (1.17 + 0.987i)7-s + (−0.341 − 0.0915i)8-s + (−0.563 + 0.976i)10-s + (0.000508 + 0.000880i)11-s + (0.411 + 0.192i)13-s + (1.04 − 0.281i)14-s + (−0.191 + 0.160i)16-s + (1.62 − 0.760i)17-s + (1.24 − 0.868i)19-s + (0.336 + 0.722i)20-s + (0.000716 + 6.26e−5i)22-s + (0.391 + 1.46i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61262 - 0.251810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61262 - 0.251810i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.573 + 0.819i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-5.96 + 1.17i)T \) |
| good | 5 | \( 1 + (3.55 - 0.310i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-3.11 - 2.61i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.00168 - 0.00292i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.48 - 0.692i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-6.72 + 3.13i)T + (10.9 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-5.40 + 3.78i)T + (6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (-1.87 - 7.01i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.451 - 1.68i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (6.34 - 6.34i)T - 31iT^{2} \) |
| 41 | \( 1 + (-8.09 + 2.94i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (3.92 + 3.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.830 + 0.479i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.38 + 8.80i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.918 - 10.4i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (4.66 - 10.0i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (-6.84 + 8.15i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.56 - 1.15i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 - 1.04iT - 73T^{2} \) |
| 79 | \( 1 + (-0.482 - 5.51i)T + (-77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 1.71i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.68 + 0.322i)T + (87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (4.41 - 1.18i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.00566119692360956415925747860, −9.563266511532744230989441371195, −8.782675806457001896794888702523, −7.76858258853806616970876318215, −7.25606870954739357198183102810, −5.48429919062652051577881150140, −5.00985413905677558138574253915, −3.73876207931447088336393250074, −2.94820125577837955238039816761, −1.24635055340664211134665605462,
1.02554368369288927283859639653, 3.38819768016837722216663620899, 4.09369289171015317234420602239, 4.89095126177585566850443972161, 6.04456826251648336166063306246, 7.40031605853243995229578401468, 7.940945925288049481035834195407, 8.154223025320508843188244271708, 9.671992945596049207183191079721, 10.89684953006661985430589930184
