| L(s) = 1 | − 2.23·5-s − 2.82i·7-s + 0.763·11-s − 5.45i·13-s − 7.40i·17-s + 19-s + 1.08i·23-s + 5.00·25-s − 4.47·29-s − 4·31-s + 6.32i·35-s + 2.62i·37-s + 6·41-s + 8.48i·43-s + 8.48i·47-s + ⋯ |
| L(s) = 1 | − 0.999·5-s − 1.06i·7-s + 0.230·11-s − 1.51i·13-s − 1.79i·17-s + 0.229·19-s + 0.225i·23-s + 1.00·25-s − 0.830·29-s − 0.718·31-s + 1.06i·35-s + 0.431i·37-s + 0.937·41-s + 1.29i·43-s + 1.23i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7036272924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7036272924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 5.45iT - 13T^{2} \) |
| 17 | \( 1 + 7.40iT - 17T^{2} \) |
| 23 | \( 1 - 1.08iT - 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2.62iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 8.48iT - 47T^{2} \) |
| 53 | \( 1 + 2.62iT - 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 5.24iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
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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
−7.87481336081253607273321849540, −7.60348619830517999059421703279, −7.03853336281000959389437370925, −5.97005063300533598044894834300, −5.01151721678310213544845064684, −4.37409949195492731957838745771, −3.41638607961360860194859210739, −2.87396856731862810178774343827, −1.12678316238531087107948979921, −0.23984122716145750447600800904,
1.57976899284297860696924778004, 2.45990072132861203717203710866, 3.79911603811935288755354628856, 4.06443273196027392325262222250, 5.22123225994634052713105223913, 5.99203922319658122254412021604, 6.77401103377938225469431183695, 7.50020851779565345339093288757, 8.364769969518123280970300600490, 8.899624525338501703162358699554
