| L(s) = 1 | + 3.70·5-s + 1.70·7-s + 1.70·11-s + 6·13-s − 3.70·17-s + 19-s − 4·23-s + 8.70·25-s − 2·29-s − 3.40·31-s + 6.29·35-s + 9.40·37-s − 9.40·41-s − 9.10·43-s − 5.70·47-s − 4.10·49-s + 6·53-s + 6.29·55-s + 4·59-s + 7.70·61-s + 22.2·65-s − 12·67-s + 0.298·73-s + 2.89·77-s + 14.8·79-s − 14.8·83-s − 13.7·85-s + ⋯ |
| L(s) = 1 | + 1.65·5-s + 0.643·7-s + 0.513·11-s + 1.66·13-s − 0.897·17-s + 0.229·19-s − 0.834·23-s + 1.74·25-s − 0.371·29-s − 0.611·31-s + 1.06·35-s + 1.54·37-s − 1.46·41-s − 1.38·43-s − 0.831·47-s − 0.586·49-s + 0.824·53-s + 0.849·55-s + 0.520·59-s + 0.986·61-s + 2.75·65-s − 1.46·67-s + 0.0349·73-s + 0.329·77-s + 1.66·79-s − 1.62·83-s − 1.48·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.599833191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599833191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 9.10T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 0.298T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.623985987723454647462811279287, −8.783401883555051252007787321317, −8.256174612138018446420011814713, −6.89665093754727264796508000560, −6.20056603293390630921616778018, −5.60394115913518302729497462908, −4.59479931383095256030422345019, −3.47298856385186742936607042384, −2.07999778394713346412751335902, −1.40012959657815816710040593728,
1.40012959657815816710040593728, 2.07999778394713346412751335902, 3.47298856385186742936607042384, 4.59479931383095256030422345019, 5.60394115913518302729497462908, 6.20056603293390630921616778018, 6.89665093754727264796508000560, 8.256174612138018446420011814713, 8.783401883555051252007787321317, 9.623985987723454647462811279287
