| L(s) = 1 | − 5-s − 11-s − 4·13-s − 3·17-s − 5·19-s + 3·23-s + 25-s + 3·29-s + 4·31-s + 11·37-s − 6·41-s + 43-s + 3·47-s − 12·53-s + 55-s − 15·59-s − 10·61-s + 4·65-s + 4·67-s − 9·71-s + 2·73-s + 16·79-s − 12·83-s + 3·85-s − 12·89-s + 5·95-s + 17·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.10·13-s − 0.727·17-s − 1.14·19-s + 0.625·23-s + 1/5·25-s + 0.557·29-s + 0.718·31-s + 1.80·37-s − 0.937·41-s + 0.152·43-s + 0.437·47-s − 1.64·53-s + 0.134·55-s − 1.95·59-s − 1.28·61-s + 0.496·65-s + 0.488·67-s − 1.06·71-s + 0.234·73-s + 1.80·79-s − 1.31·83-s + 0.325·85-s − 1.27·89-s + 0.512·95-s + 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5238983436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5238983436\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.49004255052884, −11.99500963911416, −11.64445025945692, −10.91416903225329, −10.83519144528854, −10.28515259300170, −9.671055826340600, −9.341355041294603, −8.842095612685601, −8.290974703608829, −7.887161861659206, −7.518977407951061, −6.939144998819456, −6.387695044355216, −6.203453068674970, −5.381993518610003, −4.833388532178572, −4.412647546587114, −4.250736014549742, −3.234474550278333, −2.898887232603381, −2.381409815739479, −1.775975189625134, −1.028896917928954, −0.1993884259788694,
0.1993884259788694, 1.028896917928954, 1.775975189625134, 2.381409815739479, 2.898887232603381, 3.234474550278333, 4.250736014549742, 4.412647546587114, 4.833388532178572, 5.381993518610003, 6.203453068674970, 6.387695044355216, 6.939144998819456, 7.518977407951061, 7.887161861659206, 8.290974703608829, 8.842095612685601, 9.341355041294603, 9.671055826340600, 10.28515259300170, 10.83519144528854, 10.91416903225329, 11.64445025945692, 11.99500963911416, 12.49004255052884
