| L(s) = 1 | − 2·4-s + 5-s − 3·9-s − 5·11-s + 4·16-s + 7·17-s − 2·20-s − 4·23-s − 4·25-s + 6·36-s − 43-s + 10·44-s − 3·45-s − 13·47-s − 5·55-s − 15·61-s − 8·64-s − 14·68-s + 11·73-s + 4·80-s + 9·81-s + 16·83-s + 7·85-s + 8·92-s + 15·99-s + 8·100-s + 101-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 9-s − 1.50·11-s + 16-s + 1.69·17-s − 0.447·20-s − 0.834·23-s − 4/5·25-s + 36-s − 0.152·43-s + 1.50·44-s − 0.447·45-s − 1.89·47-s − 0.674·55-s − 1.92·61-s − 64-s − 1.69·68-s + 1.28·73-s + 0.447·80-s + 81-s + 1.75·83-s + 0.759·85-s + 0.834·92-s + 1.50·99-s + 4/5·100-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7267335399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7267335399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.93338341171841, −15.05541642587174, −14.72057085053164, −13.99155090380658, −13.72045380389302, −13.24099011283463, −12.45320680140376, −12.17906525331502, −11.36405798971451, −10.65276200192603, −10.08157032935927, −9.726945326349527, −9.111383536697211, −8.267413130481259, −7.967513338280867, −7.553998775165232, −6.256118648471147, −5.857534624551940, −5.174634771223662, −4.931860283760576, −3.783979901863120, −3.238508387575103, −2.515778068275456, −1.529212341948587, −0.3625642638649373,
0.3625642638649373, 1.529212341948587, 2.515778068275456, 3.238508387575103, 3.783979901863120, 4.931860283760576, 5.174634771223662, 5.857534624551940, 6.256118648471147, 7.553998775165232, 7.967513338280867, 8.267413130481259, 9.111383536697211, 9.726945326349527, 10.08157032935927, 10.65276200192603, 11.36405798971451, 12.17906525331502, 12.45320680140376, 13.24099011283463, 13.72045380389302, 13.99155090380658, 14.72057085053164, 15.05541642587174, 15.93338341171841
