| L(s) = 1 | + 3-s + 4·5-s + 9-s + 6·11-s − 5·13-s + 4·15-s − 2·17-s − 19-s − 6·23-s + 11·25-s + 27-s + 3·31-s + 6·33-s + 3·37-s − 5·39-s + 6·41-s + 5·43-s + 4·45-s + 4·47-s − 2·51-s − 6·53-s + 24·55-s − 57-s + 6·59-s + 2·61-s − 20·65-s + 7·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 1.80·11-s − 1.38·13-s + 1.03·15-s − 0.485·17-s − 0.229·19-s − 1.25·23-s + 11/5·25-s + 0.192·27-s + 0.538·31-s + 1.04·33-s + 0.493·37-s − 0.800·39-s + 0.937·41-s + 0.762·43-s + 0.596·45-s + 0.583·47-s − 0.280·51-s − 0.824·53-s + 3.23·55-s − 0.132·57-s + 0.781·59-s + 0.256·61-s − 2.48·65-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.751665930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.751665930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.479482857768206320780868727094, −7.53368597239017508800052011504, −6.62275025072629081134542115960, −6.31674176500683569963423138421, −5.43717122117166311570606648966, −4.54931090802760388666326703236, −3.80361230583842946816101240129, −2.49601576316847006991344557543, −2.13794531072131045349400235457, −1.11848749451479838411693176323,
1.11848749451479838411693176323, 2.13794531072131045349400235457, 2.49601576316847006991344557543, 3.80361230583842946816101240129, 4.54931090802760388666326703236, 5.43717122117166311570606648966, 6.31674176500683569963423138421, 6.62275025072629081134542115960, 7.53368597239017508800052011504, 8.479482857768206320780868727094
