L(s) = 1 | − 3-s − 2·4-s − 5-s − 2·9-s + 3·11-s + 2·12-s + 15-s + 4·16-s − 3·17-s + 2·19-s + 2·20-s − 6·23-s + 25-s + 5·27-s + 3·29-s − 4·31-s − 3·33-s + 4·36-s − 2·37-s − 12·41-s − 10·43-s − 6·44-s + 2·45-s + 9·47-s − 4·48-s + 3·51-s + 12·53-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s + 0.904·11-s + 0.577·12-s + 0.258·15-s + 16-s − 0.727·17-s + 0.458·19-s + 0.447·20-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.718·31-s − 0.522·33-s + 2/3·36-s − 0.328·37-s − 1.87·41-s − 1.52·43-s − 0.904·44-s + 0.298·45-s + 1.31·47-s − 0.577·48-s + 0.420·51-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4623028957\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4623028957\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.73745345006956, −14.09541738144420, −13.73201683413446, −13.37560523200741, −12.43791849004963, −12.14829674417630, −11.70015344582317, −11.23970877266190, −10.43040736530272, −10.14004666497211, −9.373345651997256, −8.842824832479709, −8.476137813606813, −7.961529784856642, −7.128763540209931, −6.603637092018811, −5.984287675883349, −5.364246026512435, −4.910289088720229, −4.189939124829229, −3.716174145372085, −3.111054668398681, −2.079784218480817, −1.181238802550996, −0.2888610215191289,
0.2888610215191289, 1.181238802550996, 2.079784218480817, 3.111054668398681, 3.716174145372085, 4.189939124829229, 4.910289088720229, 5.364246026512435, 5.984287675883349, 6.603637092018811, 7.128763540209931, 7.961529784856642, 8.476137813606813, 8.842824832479709, 9.373345651997256, 10.14004666497211, 10.43040736530272, 11.23970877266190, 11.70015344582317, 12.14829674417630, 12.43791849004963, 13.37560523200741, 13.73201683413446, 14.09541738144420, 14.73745345006956