L(s) = 1 | − 5-s − 7-s − 11-s + 6·17-s − 6·19-s + 4·23-s + 25-s − 8·29-s − 8·31-s + 35-s + 6·37-s + 4·41-s + 12·47-s + 49-s + 6·53-s + 55-s + 4·59-s − 10·61-s + 4·67-s − 8·71-s − 4·73-s + 77-s + 2·79-s + 18·83-s − 6·85-s − 6·89-s + 6·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.45·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 1.48·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s + 0.624·41-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 0.520·59-s − 1.28·61-s + 0.488·67-s − 0.949·71-s − 0.468·73-s + 0.113·77-s + 0.225·79-s + 1.97·83-s − 0.650·85-s − 0.635·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.52296435788981, −15.99772418188439, −15.09547457143634, −14.90869549161202, −14.41290904905033, −13.47993148746966, −13.05958812899056, −12.47898433800416, −12.09481327641409, −11.20883966492993, −10.79308681814232, −10.28623852702667, −9.405029065202531, −9.071912480529059, −8.303072628603439, −7.502948684465963, −7.347961618610840, −6.369528011358267, −5.710472383720719, −5.207875420119959, −4.184386843683818, −3.749644327231435, −2.925648813752947, −2.144015782402475, −1.056444203329655, 0,
1.056444203329655, 2.144015782402475, 2.925648813752947, 3.749644327231435, 4.184386843683818, 5.207875420119959, 5.710472383720719, 6.369528011358267, 7.347961618610840, 7.502948684465963, 8.303072628603439, 9.071912480529059, 9.405029065202531, 10.28623852702667, 10.79308681814232, 11.20883966492993, 12.09481327641409, 12.47898433800416, 13.05958812899056, 13.47993148746966, 14.41290904905033, 14.90869549161202, 15.09547457143634, 15.99772418188439, 16.52296435788981