| L(s) = 1 | − 5-s − 11-s − 2·13-s − 2·17-s − 4·19-s + 8·23-s + 25-s + 2·29-s − 8·31-s + 6·37-s − 2·41-s + 4·43-s − 7·49-s − 6·53-s + 55-s + 4·59-s − 10·61-s + 2·65-s + 12·67-s + 2·73-s − 8·79-s + 12·83-s + 2·85-s + 6·89-s + 4·95-s − 14·97-s + 10·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s + 0.609·43-s − 49-s − 0.824·53-s + 0.134·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s + 1.46·67-s + 0.234·73-s − 0.900·79-s + 1.31·83-s + 0.216·85-s + 0.635·89-s + 0.410·95-s − 1.42·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321865022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321865022\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.84090724876929960381541381327, −7.12461159450538033061171951398, −6.61554538680727002200182259806, −5.73600785864903780510761394391, −4.90387030668428125493455137488, −4.42548690159715196437362679938, −3.46956241591596376423932923473, −2.72407991357128010705273762139, −1.83097473204003508583731329730, −0.55665623790118299370900042508,
0.55665623790118299370900042508, 1.83097473204003508583731329730, 2.72407991357128010705273762139, 3.46956241591596376423932923473, 4.42548690159715196437362679938, 4.90387030668428125493455137488, 5.73600785864903780510761394391, 6.61554538680727002200182259806, 7.12461159450538033061171951398, 7.84090724876929960381541381327
