L(s) = 1 | + 0.0549i·5-s + 7-s + 2.63i·11-s + 3.67i·13-s − 3.16·17-s + 3.07i·19-s + 2.86·23-s + 4.99·25-s + 10.1i·29-s − 9.32·31-s + 0.0549i·35-s − 0.774i·37-s − 6.36·41-s − 9.98i·43-s − 12.3·47-s + ⋯ |
L(s) = 1 | + 0.0245i·5-s + 0.377·7-s + 0.794i·11-s + 1.02i·13-s − 0.766·17-s + 0.706i·19-s + 0.598·23-s + 0.999·25-s + 1.89i·29-s − 1.67·31-s + 0.00928i·35-s − 0.127i·37-s − 0.993·41-s − 1.52i·43-s − 1.79·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8235977679\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8235977679\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 - 0.0549iT - 5T^{2} \) |
| 11 | \( 1 - 2.63iT - 11T^{2} \) |
| 13 | \( 1 - 3.67iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 3.07iT - 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 - 10.1iT - 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 + 0.774iT - 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 + 9.98iT - 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 3.39iT - 53T^{2} \) |
| 59 | \( 1 + 6.93iT - 59T^{2} \) |
| 61 | \( 1 - 8.35iT - 61T^{2} \) |
| 67 | \( 1 + 8.93iT - 67T^{2} \) |
| 71 | \( 1 - 4.28T + 71T^{2} \) |
| 73 | \( 1 - 8.38T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.574788397046003630326604577013, −7.54876110763442011950150837785, −6.90415579610635902759069615714, −6.55692765698626865290713156317, −5.23872542791739960039368156432, −4.96870912474217138260323122866, −3.99637315031583996782845618752, −3.27519964369773269051505457577, −2.04155863156007886665344119787, −1.52577281624139652726553959719,
0.20408788389810471514423356198, 1.28723452011757111973048910573, 2.50300254509273162789504042894, 3.16529628961934030295718698536, 4.10721521122838354931210469149, 4.96684290402380437708012803907, 5.51381807289873981687954792219, 6.39857734709132142641115971405, 6.98293900763097203116516042113, 7.964914784555984297831047783435