L(s) = 1 | + (−1 + 2i)5-s + i·7-s + 6·11-s − 2i·13-s − 2i·17-s + 4·19-s − 4i·23-s + (−3 − 4i)25-s + 2·29-s + 2·31-s + (−2 − i)35-s − 10i·37-s − 6·41-s − 2i·43-s − 2i·47-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.894i)5-s + 0.377i·7-s + 1.80·11-s − 0.554i·13-s − 0.485i·17-s + 0.917·19-s − 0.834i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s + 0.359·31-s + (−0.338 − 0.169i)35-s − 1.64i·37-s − 0.937·41-s − 0.304i·43-s − 0.291i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.892032022\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.892032022\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−8.155807575656471432744686544768, −7.36095340221376516654939139371, −6.74441008379582629140608651839, −6.20162334697453255935169739523, −5.31528950753101128554555254062, −4.34371805850461078144904530455, −3.58317391838343549071027363000, −2.93602546988935205610830885606, −1.88834890380341325451942590923, −0.59585554502381822404249494613,
1.11282886335662339463885988937, 1.54687819882890179495122354880, 3.15858759785414045170008819792, 3.92611637615677545031048206575, 4.46152639399575343392414033538, 5.27594188110129571461131574342, 6.23531266231085847618114554260, 6.83080839185318661308765045518, 7.61092480550294512553826920740, 8.347878784668809701242172101138