| L(s) = 1 | − 3-s + 4·7-s + 9-s + 6·13-s + 6·17-s + 8·19-s − 4·21-s − 27-s + 6·29-s − 6·37-s − 6·39-s + 10·41-s − 8·43-s + 9·49-s − 6·51-s − 6·53-s − 8·57-s + 4·59-s + 2·61-s + 4·63-s + 12·67-s − 8·71-s + 2·73-s + 4·79-s + 81-s − 12·83-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.66·13-s + 1.45·17-s + 1.83·19-s − 0.872·21-s − 0.192·27-s + 1.11·29-s − 0.986·37-s − 0.960·39-s + 1.56·41-s − 1.21·43-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 1.05·57-s + 0.520·59-s + 0.256·61-s + 0.503·63-s + 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.450·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.998744451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.998744451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.04739857957777, −13.82839228566730, −13.10252453654072, −12.48029566614595, −11.94210272943359, −11.57353445043560, −11.18830194248412, −10.71894406010129, −10.12126138230742, −9.676980014512737, −8.952136938557569, −8.306671300570888, −8.053125234826283, −7.464536718158062, −6.911551211981489, −6.183998339225691, −5.599805450824040, −5.265511773243743, −4.748763710009115, −3.988504911455753, −3.448412174590542, −2.804966995157879, −1.692055038520019, −1.240271917702324, −0.8179761467113128,
0.8179761467113128, 1.240271917702324, 1.692055038520019, 2.804966995157879, 3.448412174590542, 3.988504911455753, 4.748763710009115, 5.265511773243743, 5.599805450824040, 6.183998339225691, 6.911551211981489, 7.464536718158062, 8.053125234826283, 8.306671300570888, 8.952136938557569, 9.676980014512737, 10.12126138230742, 10.71894406010129, 11.18830194248412, 11.57353445043560, 11.94210272943359, 12.48029566614595, 13.10252453654072, 13.82839228566730, 14.04739857957777
