| L(s) = 1 | + 2·5-s − 11-s + 6·13-s − 2·17-s + 8·19-s + 4·23-s − 25-s − 2·29-s + 8·31-s + 6·37-s − 2·41-s + 8·43-s − 4·47-s − 2·53-s − 2·55-s − 12·59-s − 10·61-s + 12·65-s − 12·67-s + 12·71-s − 10·73-s − 8·79-s + 12·83-s − 4·85-s + 10·89-s + 16·95-s + 14·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.301·11-s + 1.66·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.583·47-s − 0.274·53-s − 0.269·55-s − 1.56·59-s − 1.28·61-s + 1.48·65-s − 1.46·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + 1.31·83-s − 0.433·85-s + 1.05·89-s + 1.64·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.763877665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.763877665\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.81135093677097, −14.07271628374106, −13.72074544773978, −13.35742994267789, −13.00232349496685, −12.14321610325124, −11.67757133452533, −11.02494550814592, −10.73181417426622, −9.996341476689533, −9.477238125587071, −9.084854484056360, −8.492012069666120, −7.769794058829883, −7.368537350441861, −6.482585722846666, −6.009735059985510, −5.715123948442000, −4.832302234238119, −4.396115457498189, −3.302200304605249, −3.092362408651668, −2.119505565500325, −1.366092473434093, −0.7791185295496685,
0.7791185295496685, 1.366092473434093, 2.119505565500325, 3.092362408651668, 3.302200304605249, 4.396115457498189, 4.832302234238119, 5.715123948442000, 6.009735059985510, 6.482585722846666, 7.368537350441861, 7.769794058829883, 8.492012069666120, 9.084854484056360, 9.477238125587071, 9.996341476689533, 10.73181417426622, 11.02494550814592, 11.67757133452533, 12.14321610325124, 13.00232349496685, 13.35742994267789, 13.72074544773978, 14.07271628374106, 14.81135093677097
