| L(s) = 1 | + (−0.707 + 2.12i)5-s + 2.82·7-s + 4.24·17-s − 4i·23-s + (−3.99 − 3i)25-s + 4.24·29-s − 8.48i·31-s + (−2.00 + 6i)35-s + 6·37-s − 4.24i·41-s − 5.65i·43-s − 4i·47-s + 1.00·49-s + 4.24i·53-s + 12i·59-s + ⋯ |
| L(s) = 1 | + (−0.316 + 0.948i)5-s + 1.06·7-s + 1.02·17-s − 0.834i·23-s + (−0.799 − 0.600i)25-s + 0.787·29-s − 1.52i·31-s + (−0.338 + 1.01i)35-s + 0.986·37-s − 0.662i·41-s − 0.862i·43-s − 0.583i·47-s + 0.142·49-s + 0.582i·53-s + 1.56i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.185869711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.185869711\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 2.12i)T \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 5.65iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
| 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
−7.85439455667584582010179816474, −7.63971438908482009429392962392, −6.68254569661503189081587677878, −6.03003588519368312924565715996, −5.20869512070133951188578082011, −4.38502388096969163717252620122, −3.67934992940711884315638663380, −2.72025137987748086078361590478, −1.97527155561328159304785324036, −0.68448656404734519856781857105,
1.01737622402434434673120839396, 1.57739327321865674109395996583, 2.86168702193150599922765552129, 3.80018469238022702204798418264, 4.63699483908939159227837482055, 5.13880158617508285231346755160, 5.78596217989301688113056473305, 6.80251651685314032528018834640, 7.66125172031772712220923181857, 8.171753994990697508151275186078
