| L(s) = 1 | − 2·5-s + 7-s − 6·13-s + 2·17-s + 4·23-s − 25-s − 10·29-s + 8·31-s − 2·35-s − 6·37-s − 2·41-s − 4·43-s − 8·47-s + 49-s − 10·53-s + 12·59-s − 2·61-s + 12·65-s − 12·67-s − 12·71-s − 14·73-s + 8·79-s − 12·83-s − 4·85-s − 2·89-s − 6·91-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.66·13-s + 0.485·17-s + 0.834·23-s − 1/5·25-s − 1.85·29-s + 1.43·31-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s + 1.56·59-s − 0.256·61-s + 1.48·65-s − 1.46·67-s − 1.42·71-s − 1.63·73-s + 0.900·79-s − 1.31·83-s − 0.433·85-s − 0.211·89-s − 0.628·91-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.58499390676019, −13.06484837075585, −12.66287110484900, −12.05586234745037, −11.78288498269713, −11.40161507087757, −10.92353317889866, −10.13885014585367, −9.973494167935938, −9.417044851517845, −8.725758122482234, −8.406239248031063, −7.691069818239150, −7.447479897689884, −7.105300342567136, −6.404817255250734, −5.760996064556869, −5.129982408049872, −4.801928464445184, −4.297694018089420, −3.616904556738721, −3.114565713492767, −2.562760969205510, −1.787683820763593, −1.233265125577972, 0, 0,
1.233265125577972, 1.787683820763593, 2.562760969205510, 3.114565713492767, 3.616904556738721, 4.297694018089420, 4.801928464445184, 5.129982408049872, 5.760996064556869, 6.404817255250734, 7.105300342567136, 7.447479897689884, 7.691069818239150, 8.406239248031063, 8.725758122482234, 9.417044851517845, 9.973494167935938, 10.13885014585367, 10.92353317889866, 11.40161507087757, 11.78288498269713, 12.05586234745037, 12.66287110484900, 13.06484837075585, 13.58499390676019
