| L(s) = 1 | − 3-s − 3·7-s + 9-s + 3·11-s − 4·13-s − 3·19-s + 3·21-s − 3·23-s − 27-s − 6·29-s + 31-s − 3·33-s − 4·37-s + 4·39-s + 6·41-s − 5·43-s + 8·47-s + 2·49-s − 53-s + 3·57-s + 14·59-s − 2·61-s − 3·63-s + 10·67-s + 3·69-s − 7·71-s + 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 0.688·19-s + 0.654·21-s − 0.625·23-s − 0.192·27-s − 1.11·29-s + 0.179·31-s − 0.522·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s − 0.762·43-s + 1.16·47-s + 2/7·49-s − 0.137·53-s + 0.397·57-s + 1.82·59-s − 0.256·61-s − 0.377·63-s + 1.22·67-s + 0.361·69-s − 0.830·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 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
−15.11844168964750, −14.63650621549781, −14.16352525822609, −13.45680348328161, −12.90485751824051, −12.54895270134139, −11.97418708538733, −11.61714175388659, −10.90562986106328, −10.28774394187104, −9.888738436037400, −9.326759894227655, −8.941248261437710, −8.095129747246592, −7.425820549229580, −6.864481676419144, −6.474702020959865, −5.865305831106030, −5.307621233411537, −4.547106545948724, −3.905204357190595, −3.437946699956301, −2.462712755584648, −1.899462113940530, −0.7734987818399738, 0,
0.7734987818399738, 1.899462113940530, 2.462712755584648, 3.437946699956301, 3.905204357190595, 4.547106545948724, 5.307621233411537, 5.865305831106030, 6.474702020959865, 6.864481676419144, 7.425820549229580, 8.095129747246592, 8.941248261437710, 9.326759894227655, 9.888738436037400, 10.28774394187104, 10.90562986106328, 11.61714175388659, 11.97418708538733, 12.54895270134139, 12.90485751824051, 13.45680348328161, 14.16352525822609, 14.63650621549781, 15.11844168964750
