| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 16-s + 6·17-s − 20-s + 4·23-s + 25-s + 10·29-s − 32-s − 6·34-s + 6·37-s + 40-s + 2·41-s − 4·43-s − 4·46-s − 7·49-s − 50-s + 6·53-s − 10·58-s + 6·61-s + 64-s − 4·67-s + 6·68-s + 16·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1/4·16-s + 1.45·17-s − 0.223·20-s + 0.834·23-s + 1/5·25-s + 1.85·29-s − 0.176·32-s − 1.02·34-s + 0.986·37-s + 0.158·40-s + 0.312·41-s − 0.609·43-s − 0.589·46-s − 49-s − 0.141·50-s + 0.824·53-s − 1.31·58-s + 0.768·61-s + 1/8·64-s − 0.488·67-s + 0.727·68-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.598043520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598043520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.18981683691414, −15.49508307659155, −15.10014052663501, −14.36187980394450, −14.04157674042167, −13.07455574509717, −12.62144660669686, −11.93473248569670, −11.59171412110304, −10.85759533527549, −10.32688176674414, −9.751899311704390, −9.248055388075145, −8.332366044662225, −8.155619386686307, −7.411538982698439, −6.797947862885944, −6.205544024803149, −5.364365361714890, −4.764822119297848, −3.843802390178085, −3.118624810964082, −2.498122448724335, −1.313707819408330, −0.6906590058118197,
0.6906590058118197, 1.313707819408330, 2.498122448724335, 3.118624810964082, 3.843802390178085, 4.764822119297848, 5.364365361714890, 6.205544024803149, 6.797947862885944, 7.411538982698439, 8.155619386686307, 8.332366044662225, 9.248055388075145, 9.751899311704390, 10.32688176674414, 10.85759533527549, 11.59171412110304, 11.93473248569670, 12.62144660669686, 13.07455574509717, 14.04157674042167, 14.36187980394450, 15.10014052663501, 15.49508307659155, 16.18981683691414
