L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 4·11-s + 2·13-s + 16-s + 6·17-s − 19-s − 2·20-s − 4·22-s + 4·23-s − 25-s − 2·26-s + 2·29-s + 4·31-s − 32-s − 6·34-s + 10·37-s + 38-s + 2·40-s − 10·41-s + 4·43-s + 4·44-s − 4·46-s + 4·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.229·19-s − 0.447·20-s − 0.852·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.162·38-s + 0.316·40-s − 1.56·41-s + 0.609·43-s + 0.603·44-s − 0.589·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9193234686\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9193234686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−11.67427667218591504436139255375, −10.56846810837709328705122029026, −9.614423289854631576305516246310, −8.666746132210989964286153997649, −7.87114970877127535090362209983, −6.93381933813641935760483898363, −5.88087715977795198376928556594, −4.27229588770101457625379469521, −3.17802027729570401593965112053, −1.15871174429474862980802238748,
1.15871174429474862980802238748, 3.17802027729570401593965112053, 4.27229588770101457625379469521, 5.88087715977795198376928556594, 6.93381933813641935760483898363, 7.87114970877127535090362209983, 8.666746132210989964286153997649, 9.614423289854631576305516246310, 10.56846810837709328705122029026, 11.67427667218591504436139255375