| L(s) = 1 | − 3·7-s − 3·11-s + 13-s + 7·17-s + 2·19-s + 23-s − 6·29-s + 10·31-s + 37-s + 41-s − 4·43-s − 4·47-s + 2·49-s − 53-s + 13·61-s − 8·67-s + 5·71-s + 10·73-s + 9·77-s + 79-s − 6·83-s − 9·89-s − 3·91-s − 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.904·11-s + 0.277·13-s + 1.69·17-s + 0.458·19-s + 0.208·23-s − 1.11·29-s + 1.79·31-s + 0.164·37-s + 0.156·41-s − 0.609·43-s − 0.583·47-s + 2/7·49-s − 0.137·53-s + 1.66·61-s − 0.977·67-s + 0.593·71-s + 1.17·73-s + 1.02·77-s + 0.112·79-s − 0.658·83-s − 0.953·89-s − 0.314·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.07183586060835, −13.43467903851560, −13.10017719570418, −12.71067941542514, −12.09748041779314, −11.74127738516546, −11.09265436117007, −10.51654340336640, −9.985688295160043, −9.720628730349367, −9.299201332894781, −8.407769725483267, −8.088092833121331, −7.560939187921375, −6.965215741730072, −6.429468334904792, −5.893570386630613, −5.329139514445612, −4.968986709102954, −3.992543229638699, −3.552201543624424, −2.921308256831546, −2.586083739428018, −1.523515274673647, −0.8429824176093798, 0,
0.8429824176093798, 1.523515274673647, 2.586083739428018, 2.921308256831546, 3.552201543624424, 3.992543229638699, 4.968986709102954, 5.329139514445612, 5.893570386630613, 6.429468334904792, 6.965215741730072, 7.560939187921375, 8.088092833121331, 8.407769725483267, 9.299201332894781, 9.720628730349367, 9.985688295160043, 10.51654340336640, 11.09265436117007, 11.74127738516546, 12.09748041779314, 12.71067941542514, 13.10017719570418, 13.43467903851560, 14.07183586060835
