| L(s) = 1 | − 1.41i·3-s + 1.41i·5-s − 2·7-s + 0.999·9-s − 1.41i·11-s − 1.41i·13-s + 2.00·15-s + 2·17-s − 4.24i·19-s + 2.82i·21-s − 6·23-s + 2.99·25-s − 5.65i·27-s − 4.24i·29-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + 0.632i·5-s − 0.755·7-s + 0.333·9-s − 0.426i·11-s − 0.392i·13-s + 0.516·15-s + 0.485·17-s − 0.973i·19-s + 0.617i·21-s − 1.25·23-s + 0.599·25-s − 1.08i·27-s − 0.787i·29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.360380859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.360380859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 7.07iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 4.24iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 7.07iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.41iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−9.948975380079506745966139116837, −8.832670494873253177321195957629, −7.899411765043100827645955048993, −7.14775107391726688352507500281, −6.46078746563502339471693998299, −5.74531690973983054508349506842, −4.33846451309494687477940766460, −3.19994371502056489193105609228, −2.27686264849340236244693375021, −0.66311245126504812302628570019,
1.42605941539100853309724913155, 3.04721980727381412786029043622, 4.10166462469460812271823003049, 4.72812929748733416768286989422, 5.80543070102939722977492335971, 6.70344858586590378313709496923, 7.74883236945244734105833108280, 8.655161200073585778377237657365, 9.547914153327182136731585318217, 9.992676442534183611680202937412
