L(s) = 1 | + (0.707 + 1.58i)3-s + 2.23i·5-s − 7-s + (−2.00 + 2.23i)9-s + 2.23i·11-s + 3.16i·13-s + (−3.53 + 1.58i)15-s − 6.70i·17-s + (−3 + 3.16i)19-s + (−0.707 − 1.58i)21-s − 4.47i·23-s + (−4.94 − 1.58i)27-s − 5.65·29-s + 3.16i·31-s + (−3.53 + 1.58i)33-s + ⋯ |
L(s) = 1 | + (0.408 + 0.912i)3-s + 0.999i·5-s − 0.377·7-s + (−0.666 + 0.745i)9-s + 0.674i·11-s + 0.877i·13-s + (−0.912 + 0.408i)15-s − 1.62i·17-s + (−0.688 + 0.725i)19-s + (−0.154 − 0.345i)21-s − 0.932i·23-s + (−0.952 − 0.304i)27-s − 1.05·29-s + 0.567i·31-s + (−0.615 + 0.275i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.217721 + 1.27394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.217721 + 1.27394i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 1.58i)T \) |
| 19 | \( 1 + (3 - 3.16i)T \) |
good | 5 | \( 1 - 2.23iT - 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2.23iT - 11T^{2} \) |
| 13 | \( 1 - 3.16iT - 13T^{2} \) |
| 17 | \( 1 + 6.70iT - 17T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 3.16iT - 31T^{2} \) |
| 37 | \( 1 - 9.48iT - 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 2.23iT - 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 6.32iT - 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 3.16iT - 97T^{2} \) |
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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
−10.39813385594226099478384383132, −9.616180599234108856807964177736, −9.091253803798652752921299459683, −7.946265843822307648787595259390, −7.03478316735657042958477977067, −6.29168550720794500664621359888, −4.99525727939636799975678945713, −4.17101021529907217688289906249, −3.10501960608518363710646247425, −2.27218448789381555023435913855,
0.56269559371047578467566630843, 1.85105490044898194042503456436, 3.17379438390478724855635606705, 4.19514958909161710162985748078, 5.68984777965860849972952620676, 6.06644051692345301172136881689, 7.38086062968160618937879203311, 8.053999243532492629827665975352, 8.844966891534763731447996480246, 9.358581819353955272861639438178