| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s − 2·9-s + 3·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s − 3·17-s + 2·19-s + 2·20-s − 21-s + 6·23-s + 25-s + 5·27-s − 2·28-s − 3·29-s − 3·33-s − 35-s + 4·36-s − 2·37-s + 5·39-s − 12·41-s + 10·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s − 0.727·17-s + 0.458·19-s + 0.447·20-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.377·28-s − 0.557·29-s − 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s + 0.800·39-s − 1.87·41-s + 1.52·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.20170898618481, −14.57510144955188, −14.32017023845852, −13.76668016482181, −13.14590541481299, −12.45821777778913, −12.12809012613164, −11.63565191426304, −11.04405876414593, −10.59422274501715, −9.848305938485446, −9.202646778303202, −8.926174759869758, −8.365302518926808, −7.533875624733144, −7.213596363935943, −6.412135705243611, −5.756793863838243, −5.047407664470216, −4.785033194685816, −4.155849758818111, −3.360938239625395, −2.743499524316247, −1.670038737944617, −0.7444006095250255, 0,
0.7444006095250255, 1.670038737944617, 2.743499524316247, 3.360938239625395, 4.155849758818111, 4.785033194685816, 5.047407664470216, 5.756793863838243, 6.412135705243611, 7.213596363935943, 7.533875624733144, 8.365302518926808, 8.926174759869758, 9.202646778303202, 9.848305938485446, 10.59422274501715, 11.04405876414593, 11.63565191426304, 12.12809012613164, 12.45821777778913, 13.14590541481299, 13.76668016482181, 14.32017023845852, 14.57510144955188, 15.20170898618481
