L(s) = 1 | − 2-s − 4-s − 5-s + 2·7-s + 3·8-s + 10-s − 2·14-s − 16-s + 2·17-s + 2·19-s + 20-s − 8·23-s + 25-s − 2·28-s − 2·29-s − 2·31-s − 5·32-s − 2·34-s − 2·35-s + 8·37-s − 2·38-s − 3·40-s + 2·41-s + 4·43-s + 8·46-s − 4·47-s − 3·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.755·7-s + 1.06·8-s + 0.316·10-s − 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.458·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.377·28-s − 0.371·29-s − 0.359·31-s − 0.883·32-s − 0.342·34-s − 0.338·35-s + 1.31·37-s − 0.324·38-s − 0.474·40-s + 0.312·41-s + 0.609·43-s + 1.17·46-s − 0.583·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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
−7.68233340286501215682483919765, −7.26508993287830113349062496411, −6.05232824006697483877381014694, −5.45810447267116591022295704733, −4.49457719416755052186474250753, −4.15374103152959916326914373310, −3.14855395892641971223047474590, −1.95581807135501671067905017905, −1.13059496967601832587580884276, 0,
1.13059496967601832587580884276, 1.95581807135501671067905017905, 3.14855395892641971223047474590, 4.15374103152959916326914373310, 4.49457719416755052186474250753, 5.45810447267116591022295704733, 6.05232824006697483877381014694, 7.26508993287830113349062496411, 7.68233340286501215682483919765