| L(s) = 1 | − 0.517i·2-s + 1.73·4-s + 5-s + (1 + 2.44i)7-s − 1.93i·8-s − 0.517i·10-s + 0.378i·11-s + 1.79i·13-s + (1.26 − 0.517i)14-s + 2.46·16-s − 3.46·17-s − 1.79i·19-s + 1.73·20-s + 0.196·22-s − 1.41i·23-s + ⋯ |
| L(s) = 1 | − 0.366i·2-s + 0.866·4-s + 0.447·5-s + (0.377 + 0.925i)7-s − 0.683i·8-s − 0.163i·10-s + 0.114i·11-s + 0.497i·13-s + (0.338 − 0.138i)14-s + 0.616·16-s − 0.840·17-s − 0.411i·19-s + 0.387·20-s + 0.0418·22-s − 0.294i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.71077 - 0.195776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71077 - 0.195776i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
| good | 2 | \( 1 + 0.517iT - 2T^{2} \) |
| 11 | \( 1 - 0.378iT - 11T^{2} \) |
| 13 | \( 1 - 1.79iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 1.79iT - 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 6.69iT - 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 13.3iT - 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 15.0iT - 71T^{2} \) |
| 73 | \( 1 + 1.79iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 15.1iT - 97T^{2} \) |
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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
−11.49475427029253547432643248691, −10.96028401171689032418018008292, −9.780549594312764463730568356401, −8.977004752650543629054381254197, −7.79442081431534446587376304985, −6.62681788617958233155747121512, −5.84476632100703426418965401639, −4.48060314309832582078767236639, −2.78724457082843949553297470231, −1.85199912483380229660003245258,
1.65645164193364391998557010203, 3.20171324690582010984728758088, 4.76529562969078925914512920827, 5.93738981558368653400554332536, 6.88171267493195176981269538222, 7.69874870166444998219624472639, 8.688618764183620289780637067337, 10.06683433952854853202102205342, 10.76093906037350975238039494188, 11.50706338509626764500164257761
