| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 11-s + 15-s − 17-s + 7·19-s + 2·21-s + 25-s + 27-s − 5·29-s + 33-s + 2·35-s − 2·37-s + 5·41-s − 5·43-s + 45-s + 2·47-s − 3·49-s − 51-s − 6·53-s + 55-s + 7·57-s − 6·59-s + 14·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.258·15-s − 0.242·17-s + 1.60·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s − 0.928·29-s + 0.174·33-s + 0.338·35-s − 0.328·37-s + 0.780·41-s − 0.762·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s − 0.140·51-s − 0.824·53-s + 0.134·55-s + 0.927·57-s − 0.781·59-s + 1.79·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.879386307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.879386307\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.60395015403561, −12.07260991109317, −11.66195038032168, −11.16214046607595, −10.84633915350384, −10.22259727036117, −9.634100642197872, −9.463767028289319, −8.985172441661332, −8.390252751552469, −7.994944061026668, −7.573134768879373, −7.034206325833290, −6.669441746702943, −5.932683366818388, −5.494813043890736, −5.043967063755072, −4.538972366061333, −3.944261963755857, −3.398264843223771, −2.928796619401253, −2.276842217097084, −1.696622979975165, −1.310533824777825, −0.5489329973145713,
0.5489329973145713, 1.310533824777825, 1.696622979975165, 2.276842217097084, 2.928796619401253, 3.398264843223771, 3.944261963755857, 4.538972366061333, 5.043967063755072, 5.494813043890736, 5.932683366818388, 6.669441746702943, 7.034206325833290, 7.573134768879373, 7.994944061026668, 8.390252751552469, 8.985172441661332, 9.463767028289319, 9.634100642197872, 10.22259727036117, 10.84633915350384, 11.16214046607595, 11.66195038032168, 12.07260991109317, 12.60395015403561
