| L(s) = 1 | − 2·3-s − 2·4-s + 5-s − 7-s + 9-s − 4·11-s + 4·12-s − 2·13-s − 2·15-s + 4·16-s − 6·17-s + 5·19-s − 2·20-s + 2·21-s − 8·23-s + 25-s + 4·27-s + 2·28-s + 5·29-s − 5·31-s + 8·33-s − 35-s − 2·36-s + 2·37-s + 4·39-s + 5·41-s − 2·43-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.15·12-s − 0.554·13-s − 0.516·15-s + 16-s − 1.45·17-s + 1.14·19-s − 0.447·20-s + 0.436·21-s − 1.66·23-s + 1/5·25-s + 0.769·27-s + 0.377·28-s + 0.928·29-s − 0.898·31-s + 1.39·33-s − 0.169·35-s − 1/3·36-s + 0.328·37-s + 0.640·39-s + 0.780·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.18790675301199, −13.71096089689799, −13.05980713128891, −12.99596419439180, −12.29847192081626, −11.87669874027752, −11.34216571233619, −10.73213727784305, −10.33313647978275, −9.829746106141939, −9.522493614993016, −8.826388050614691, −8.326895956325045, −7.726841209555182, −7.246869283457789, −6.370107642271709, −6.152701810661228, −5.489328339909388, −5.060616427180991, −4.704242229342770, −4.080864286701814, −3.266066224752008, −2.644881295693424, −1.936243667709953, −0.9791361205212013, 0, 0,
0.9791361205212013, 1.936243667709953, 2.644881295693424, 3.266066224752008, 4.080864286701814, 4.704242229342770, 5.060616427180991, 5.489328339909388, 6.152701810661228, 6.370107642271709, 7.246869283457789, 7.726841209555182, 8.326895956325045, 8.826388050614691, 9.522493614993016, 9.829746106141939, 10.33313647978275, 10.73213727784305, 11.34216571233619, 11.87669874027752, 12.29847192081626, 12.99596419439180, 13.05980713128891, 13.71096089689799, 14.18790675301199
