L(s) = 1 | − 1.41·3-s + 4.24i·7-s − 0.999·9-s − 5.65i·11-s + 2·13-s − 6i·17-s + 2.82i·19-s − 6i·21-s + 7.07i·23-s + 5.65·27-s + 4i·29-s − 2.82·31-s + 8.00i·33-s − 2·37-s − 2.82·39-s + ⋯ |
L(s) = 1 | − 0.816·3-s + 1.60i·7-s − 0.333·9-s − 1.70i·11-s + 0.554·13-s − 1.45i·17-s + 0.648i·19-s − 1.30i·21-s + 1.47i·23-s + 1.08·27-s + 0.742i·29-s − 0.508·31-s + 1.39i·33-s − 0.328·37-s − 0.452·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4869316889\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4869316889\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 2.82iT - 59T^{2} \) |
| 61 | \( 1 + 14iT - 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.649424890204372987493850996120, −7.80894344305084365655293600751, −6.67888599980097360571032605997, −5.93614615088603827994714754930, −5.51264414363105305609989938263, −5.01886389728478838883597292557, −3.39337736488481790835439396310, −2.98411079846469218464920546331, −1.61242989944609127138176772612, −0.19137264158920573526259105411,
1.06081264409486018769888146154, 2.18909969233689856357187558890, 3.60121403605444803516686156994, 4.37168693396198475769701240286, 4.88377106921345217558759288056, 6.00957870421387203240410871374, 6.69668986119352689197136404833, 7.18485658499898357302945458059, 8.082869806567169440626694728201, 8.808229325939616508575549077939