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
−9.358581819353955272861639438178, −8.844966891534763731447996480246, −8.053999243532492629827665975352, −7.38086062968160618937879203311, −6.06644051692345301172136881689, −5.68984777965860849972952620676, −4.19514958909161710162985748078, −3.17379438390478724855635606705, −1.85105490044898194042503456436, −0.56269559371047578467566630843,
2.27218448789381555023435913855, 3.10501960608518363710646247425, 4.17101021529907217688289906249, 4.99525727939636799975678945713, 6.29168550720794500664621359888, 7.03478316735657042958477977067, 7.946265843822307648787595259390, 9.091253803798652752921299459683, 9.616180599234108856807964177736, 10.39813385594226099478384383132