| L(s) = 1 | − 2·5-s − 7-s + 6·13-s − 17-s + 8·23-s − 25-s − 6·29-s − 8·31-s + 2·35-s − 10·37-s + 6·41-s − 12·43-s + 49-s − 10·53-s − 8·59-s − 6·61-s − 12·65-s − 12·67-s − 6·73-s − 8·79-s + 16·83-s + 2·85-s − 2·89-s − 6·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.66·13-s − 0.242·17-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s + 0.937·41-s − 1.82·43-s + 1/7·49-s − 1.37·53-s − 1.04·59-s − 0.768·61-s − 1.48·65-s − 1.46·67-s − 0.702·73-s − 0.900·79-s + 1.75·83-s + 0.216·85-s − 0.211·89-s − 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9235043245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9235043245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.09335388534929, −13.58347476143207, −13.11410578413344, −12.76102469550621, −12.17361351648651, −11.50216914962574, −11.16204317091138, −10.79816329605985, −10.26489529277113, −9.398050532906024, −8.981887672179233, −8.657411750583466, −7.995395982787787, −7.312637612065898, −7.126336712125865, −6.214694994207655, −5.961565238371375, −5.107753089486085, −4.622803611842555, −3.787976848584128, −3.439700058217476, −3.065364340000740, −1.851713470316598, −1.386097745818580, −0.3262037150475896,
0.3262037150475896, 1.386097745818580, 1.851713470316598, 3.065364340000740, 3.439700058217476, 3.787976848584128, 4.622803611842555, 5.107753089486085, 5.961565238371375, 6.214694994207655, 7.126336712125865, 7.312637612065898, 7.995395982787787, 8.657411750583466, 8.981887672179233, 9.398050532906024, 10.26489529277113, 10.79816329605985, 11.16204317091138, 11.50216914962574, 12.17361351648651, 12.76102469550621, 13.11410578413344, 13.58347476143207, 14.09335388534929
