| L(s) = 1 | − 3·7-s + 3·11-s − 13-s − 17-s − 19-s − 4·23-s − 5·29-s + 5·37-s − 5·41-s − 8·43-s + 5·47-s + 2·49-s − 3·53-s + 6·59-s + 4·61-s + 12·67-s − 17·73-s − 9·77-s + 4·79-s − 16·83-s − 18·89-s + 3·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.904·11-s − 0.277·13-s − 0.242·17-s − 0.229·19-s − 0.834·23-s − 0.928·29-s + 0.821·37-s − 0.780·41-s − 1.21·43-s + 0.729·47-s + 2/7·49-s − 0.412·53-s + 0.781·59-s + 0.512·61-s + 1.46·67-s − 1.98·73-s − 1.02·77-s + 0.450·79-s − 1.75·83-s − 1.90·89-s + 0.314·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.88584559178515, −12.50126585187427, −11.98746630834381, −11.48712918454160, −11.27928835966607, −10.55445594849231, −10.01454651333364, −9.796042849998644, −9.355483433506393, −8.894810683697499, −8.332749275600584, −8.008566282430399, −7.165018202709155, −6.959767686709142, −6.501957249781917, −5.981635169693371, −5.611486165134009, −4.980834176296871, −4.334232298777299, −3.793693842762618, −3.629246974805066, −2.759802699390878, −2.457736163613853, −1.623523772053044, −1.168922493420474, 0, 0,
1.168922493420474, 1.623523772053044, 2.457736163613853, 2.759802699390878, 3.629246974805066, 3.793693842762618, 4.334232298777299, 4.980834176296871, 5.611486165134009, 5.981635169693371, 6.501957249781917, 6.959767686709142, 7.165018202709155, 8.008566282430399, 8.332749275600584, 8.894810683697499, 9.355483433506393, 9.796042849998644, 10.01454651333364, 10.55445594849231, 11.27928835966607, 11.48712918454160, 11.98746630834381, 12.50126585187427, 12.88584559178515
