| L(s) = 1 | + (−0.142 + 0.989i)3-s + (2.48 − 0.729i)5-s + (3.07 + 3.55i)7-s + (−0.959 − 0.281i)9-s + (0.458 + 0.294i)11-s + (−2.10 + 2.43i)13-s + (0.368 + 2.56i)15-s + (−2.87 − 6.29i)17-s + (1.04 − 2.29i)19-s + (−3.95 + 2.54i)21-s + (3.54 + 3.23i)23-s + (1.43 − 0.920i)25-s + (0.415 − 0.909i)27-s + (2.25 + 4.94i)29-s + (−0.502 − 3.49i)31-s + ⋯ |
| L(s) = 1 | + (−0.0821 + 0.571i)3-s + (1.11 − 0.326i)5-s + (1.16 + 1.34i)7-s + (−0.319 − 0.0939i)9-s + (0.138 + 0.0888i)11-s + (−0.584 + 0.675i)13-s + (0.0951 + 0.661i)15-s + (−0.697 − 1.52i)17-s + (0.240 − 0.526i)19-s + (−0.862 + 0.554i)21-s + (0.738 + 0.674i)23-s + (0.286 − 0.184i)25-s + (0.0799 − 0.175i)27-s + (0.419 + 0.917i)29-s + (−0.0901 − 0.627i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63504 + 0.823281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63504 + 0.823281i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-3.54 - 3.23i)T \) |
| good | 5 | \( 1 + (-2.48 + 0.729i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-3.07 - 3.55i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.458 - 0.294i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.10 - 2.43i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.87 + 6.29i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.04 + 2.29i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.25 - 4.94i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.502 + 3.49i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.55 + 0.455i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.29 + 0.966i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.28 - 8.94i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 3.27T + 47T^{2} \) |
| 53 | \( 1 + (0.421 + 0.486i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.02 + 6.95i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.39 + 9.70i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (10.8 - 6.99i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-6.22 + 3.99i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.14 + 11.2i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.232 - 0.268i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (5.71 + 1.67i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.95 + 13.5i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (7.28 - 2.13i)T + (81.6 - 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.16770913908017431675583414035, −9.664717650139658460544679692782, −9.302693135554399399636608684636, −8.624148159415743244429377211549, −7.31236458348439539832064650740, −6.11273336801719052870537028257, −5.00775608457564633994644985397, −4.86966050810290474628073692232, −2.79317454429737424620243112845, −1.80669001242450260031007717787,
1.23961458107679158589320193978, 2.33621843071064164460167711187, 3.98749814264766267510745451618, 5.13827971789815619398809762429, 6.16455030266352668392368538709, 7.03288005765570192992435507801, 7.910896646534615054320903607178, 8.716454708501842336080863464360, 10.21397710586657689077555312768, 10.44541511770795469635679238178
