| L(s) = 1 | − 4.46i·5-s − 0.418·7-s + 3.70i·11-s + 7.88i·17-s + 4.35·19-s + 8.99i·23-s − 14.9·25-s + 1.87i·35-s − 11.8·43-s + 7.57i·47-s − 6.82·49-s + 16.5·55-s + 10.8·61-s + 5.82·73-s − 1.54i·77-s + ⋯ |
| L(s) = 1 | − 1.99i·5-s − 0.158·7-s + 1.11i·11-s + 1.91i·17-s + 1.00·19-s + 1.87i·23-s − 2.99·25-s + 0.316i·35-s − 1.80·43-s + 1.10i·47-s − 0.974·49-s + 2.23·55-s + 1.38·61-s + 0.681·73-s − 0.176i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.226688759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226688759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.35T \) |
| good | 5 | \( 1 + 4.46iT - 5T^{2} \) |
| 7 | \( 1 + 0.418T + 7T^{2} \) |
| 11 | \( 1 - 3.70iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.88iT - 17T^{2} \) |
| 23 | \( 1 - 8.99iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 7.57iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| 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
−8.977247458881882191538186958515, −8.117168184616812806271723877211, −7.76067595918947017007789890825, −6.61110710563481539913409137871, −5.56769589324513074941397589343, −5.14232557810714496762520112949, −4.24431900345468900354665088826, −3.56152556315113691443071490765, −1.82173209835503165593003817328, −1.28837201177492236081854725319,
0.40813769632161091863707039184, 2.30402878489543813586160321173, 3.05466612529565233358387486831, 3.50452761164406480218324392579, 4.85363412100899083832799031149, 5.79375077825386609536358310006, 6.65272644032578776962287050468, 6.97376103894362629927166314785, 7.82196035051745691121291737790, 8.652716157670002335782209851025
