| L(s) = 1 | − 2.17i·5-s + 1.38·7-s + 2.59i·11-s − 1.61i·13-s + 4.65i·17-s + (−4.02 + 1.66i)19-s − 0.478i·23-s + 0.266·25-s + 6.70·29-s + 6.17i·31-s − 3.00i·35-s + 4.26i·37-s + 2.95·41-s − 11.0·43-s − 3.48i·47-s + ⋯ |
| L(s) = 1 | − 0.973i·5-s + 0.522·7-s + 0.781i·11-s − 0.446i·13-s + 1.12i·17-s + (−0.924 + 0.381i)19-s − 0.0996i·23-s + 0.0532·25-s + 1.24·29-s + 1.10i·31-s − 0.508i·35-s + 0.701i·37-s + 0.461·41-s − 1.67·43-s − 0.508i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387797819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387797819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4.02 - 1.66i)T \) |
| good | 5 | \( 1 + 2.17iT - 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 2.59iT - 11T^{2} \) |
| 13 | \( 1 + 1.61iT - 13T^{2} \) |
| 17 | \( 1 - 4.65iT - 17T^{2} \) |
| 23 | \( 1 + 0.478iT - 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 6.17iT - 31T^{2} \) |
| 37 | \( 1 - 4.26iT - 37T^{2} \) |
| 41 | \( 1 - 2.95T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 3.48iT - 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 + 8.59T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 - 13.2iT - 67T^{2} \) |
| 71 | \( 1 + 0.0858T + 71T^{2} \) |
| 73 | \( 1 + 0.467T + 73T^{2} \) |
| 79 | \( 1 - 15.5iT - 79T^{2} \) |
| 83 | \( 1 + 2.37iT - 83T^{2} \) |
| 89 | \( 1 + 5.23T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
| 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
−8.481539598253092820562350999509, −7.81921208966595416707918055147, −6.81323002761187804525888058809, −6.21406009680593368584138114061, −5.21039762554749828981744290749, −4.73282426812396202674071850257, −4.07535107375122331575523485249, −3.01255561518302845049726926245, −1.84671553541015416092513239691, −1.19166484275128914196047838589,
0.36863584848942624170521900591, 1.77372057984537644390632470718, 2.73877002865486072218725898224, 3.31295210797206601020355005960, 4.43392233422838843217108401372, 4.97055890617418246787385804642, 6.11550387461061689907399974880, 6.49022508017000123919540512163, 7.30937142647910908437122797292, 7.929294374588899867506530702981
