| L(s) = 1 | + 0.841·5-s + (−1.16 − 2.37i)7-s + 3.64i·11-s + 5.53i·13-s + 0.841·17-s − 3.06i·19-s − 3.64i·23-s − 4.29·25-s + 8.89i·29-s + 7.82i·31-s + (−0.979 − 1.99i)35-s − 3.29·37-s + 8.66·41-s + 2.32·43-s − 9.10·47-s + ⋯ |
| L(s) = 1 | + 0.376·5-s + (−0.439 − 0.898i)7-s + 1.09i·11-s + 1.53i·13-s + 0.204·17-s − 0.704i·19-s − 0.760i·23-s − 0.858·25-s + 1.65i·29-s + 1.40i·31-s + (−0.165 − 0.338i)35-s − 0.541·37-s + 1.35·41-s + 0.354·43-s − 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359191368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359191368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.16 + 2.37i)T \) |
| good | 5 | \( 1 - 0.841T + 5T^{2} \) |
| 11 | \( 1 - 3.64iT - 11T^{2} \) |
| 13 | \( 1 - 5.53iT - 13T^{2} \) |
| 17 | \( 1 - 0.841T + 17T^{2} \) |
| 19 | \( 1 + 3.06iT - 19T^{2} \) |
| 23 | \( 1 + 3.64iT - 23T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 - 7.82iT - 31T^{2} \) |
| 37 | \( 1 + 3.29T + 37T^{2} \) |
| 41 | \( 1 - 8.66T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 + 9.10T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 9.10iT - 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 0.354iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 9.10T + 83T^{2} \) |
| 89 | \( 1 - 6.97T + 89T^{2} \) |
| 97 | \( 1 - 3.57iT - 97T^{2} \) |
| 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
−9.331562268271660469581975225113, −8.757209377652641600081286650780, −7.54181387660786768746231622626, −6.88318847750326351551786795440, −6.48965472715811056639034507603, −5.15769241287806536182965613757, −4.45093028219299866927633279562, −3.62208501470239630022289818771, −2.36555299476397559570311108495, −1.33811711011497656515494907663,
0.49464789299546347055546725084, 2.09206622940068322181446631954, 3.05153112591558135982263491097, 3.81758863923969716015672707173, 5.23648070060724006108313154856, 5.98547253508658425622895496878, 6.11912136247903607262910026765, 7.81872151618426231066528871982, 7.972347928811124231729956252878, 9.071402033650443870230366904470
