| L(s) = 1 | − 2·5-s − 11-s + 4·13-s + 2·17-s + 2·19-s − 8·23-s − 25-s + 6·29-s − 6·31-s + 2·37-s − 6·41-s + 4·43-s + 6·47-s + 14·53-s + 2·55-s − 8·59-s + 4·61-s − 8·65-s + 8·67-s − 16·71-s + 14·73-s − 4·79-s − 14·83-s − 4·85-s − 12·89-s − 4·95-s − 16·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.301·11-s + 1.10·13-s + 0.485·17-s + 0.458·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.07·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 0.875·47-s + 1.92·53-s + 0.269·55-s − 1.04·59-s + 0.512·61-s − 0.992·65-s + 0.977·67-s − 1.89·71-s + 1.63·73-s − 0.450·79-s − 1.53·83-s − 0.433·85-s − 1.27·89-s − 0.410·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.28271061358949, −14.49151474100587, −13.99741394049233, −13.64730480244822, −12.96532694747848, −12.37618568298451, −11.89541270237327, −11.54322302434427, −10.90481172831054, −10.35735062961840, −9.908298547495290, −9.190686297670752, −8.494221166459999, −8.183667269240254, −7.610655102205576, −7.088905887221475, −6.379344933603290, −5.685960749890384, −5.365208149535944, −4.218985614598648, −4.054504968950065, −3.360501054619457, −2.643641870826612, −1.750963795755243, −0.9253537924072789, 0,
0.9253537924072789, 1.750963795755243, 2.643641870826612, 3.360501054619457, 4.054504968950065, 4.218985614598648, 5.365208149535944, 5.685960749890384, 6.379344933603290, 7.088905887221475, 7.610655102205576, 8.183667269240254, 8.494221166459999, 9.190686297670752, 9.908298547495290, 10.35735062961840, 10.90481172831054, 11.54322302434427, 11.89541270237327, 12.37618568298451, 12.96532694747848, 13.64730480244822, 13.99741394049233, 14.49151474100587, 15.28271061358949
