L(s) = 1 | + (0.939 − 0.342i)2-s + (0.441 − 1.67i)3-s + (0.766 − 0.642i)4-s + (2.80 + 0.493i)5-s + (−0.158 − 1.72i)6-s + (0.343 − 0.595i)7-s + (0.500 − 0.866i)8-s + (−2.61 − 1.47i)9-s + (2.80 − 0.493i)10-s + 4.77i·11-s + (−0.738 − 1.56i)12-s + (−1.94 + 0.342i)13-s + (0.119 − 0.677i)14-s + (2.06 − 4.47i)15-s + (0.173 − 0.984i)16-s + (−1.42 − 0.251i)17-s + ⋯ |
L(s) = 1 | + (0.664 − 0.241i)2-s + (0.254 − 0.967i)3-s + (0.383 − 0.321i)4-s + (1.25 + 0.220i)5-s + (−0.0646 − 0.704i)6-s + (0.130 − 0.225i)7-s + (0.176 − 0.306i)8-s + (−0.870 − 0.492i)9-s + (0.885 − 0.156i)10-s + 1.44i·11-s + (−0.213 − 0.452i)12-s + (−0.538 + 0.0948i)13-s + (0.0319 − 0.181i)14-s + (0.532 − 1.15i)15-s + (0.0434 − 0.246i)16-s + (−0.345 − 0.0610i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02262 - 1.20911i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02262 - 1.20911i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.441 + 1.67i)T \) |
| 19 | \( 1 + (1.55 - 4.07i)T \) |
good | 5 | \( 1 + (-2.80 - 0.493i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.343 + 0.595i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 4.77iT - 11T^{2} \) |
| 13 | \( 1 + (1.94 - 0.342i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.42 + 0.251i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (3.50 + 4.17i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.60 + 2.18i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + 4.18iT - 31T^{2} \) |
| 37 | \( 1 - 9.14iT - 37T^{2} \) |
| 41 | \( 1 + (-4.53 + 1.65i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (8.90 + 7.47i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (6.84 + 8.15i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-8.96 - 3.26i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-5.84 - 4.90i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.47 - 8.34i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.91 - 8.01i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.24 + 1.90i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.37 - 1.99i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.18 - 0.386i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (8.42 + 4.86i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.30 + 5.28i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (2.69 + 7.39i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−11.77236468043137149779610069608, −10.25268722855151332576858048024, −9.863220354431420100307089835566, −8.474657584161253498352286453811, −7.24661719089861852640002319712, −6.51056862421958483673971003581, −5.59107366029360538012521805426, −4.28895945467187891577555293728, −2.49988850988958903086154631630, −1.81459455355844035208841664532,
2.33053698502517367926619374228, 3.49182484962223409424991491133, 4.90628209901238390611428075031, 5.58310123026900697043794754586, 6.47717847257285629147788366128, 8.108552227816650156068178277061, 8.988366269200523117762618643978, 9.776476086486381851289606823240, 10.81942716211330044809829152318, 11.51848392485598413303244959366