L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 2·7-s + 8-s + 9-s − 2·10-s − 11-s − 12-s − 6·13-s − 2·14-s + 2·15-s + 16-s + 2·17-s + 18-s − 4·19-s − 2·20-s + 2·21-s − 22-s − 6·23-s − 24-s − 25-s − 6·26-s − 27-s − 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.29824712737608, −13.53753540523271, −12.91162442531369, −12.53042076211642, −12.23168205368991, −11.89586888501339, −11.11433401223743, −10.89171156974333, −10.21133374776764, −9.609839530967740, −9.453197663623374, −8.400617119898185, −7.823603619546646, −7.472378746094639, −7.048736056820567, −6.298430969769566, −5.925754250242193, −5.339836303763966, −4.734916860559394, −4.154572685007180, −3.818338686708138, −3.075029460047010, −2.386746421710074, −1.876237561291975, −0.6012051868632244, 0,
0.6012051868632244, 1.876237561291975, 2.386746421710074, 3.075029460047010, 3.818338686708138, 4.154572685007180, 4.734916860559394, 5.339836303763966, 5.925754250242193, 6.298430969769566, 7.048736056820567, 7.472378746094639, 7.823603619546646, 8.400617119898185, 9.453197663623374, 9.609839530967740, 10.21133374776764, 10.89171156974333, 11.11433401223743, 11.89586888501339, 12.23168205368991, 12.53042076211642, 12.91162442531369, 13.53753540523271, 14.29824712737608