L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 3·11-s − 4·13-s + 4·14-s + 16-s + 3·17-s + 5·19-s + 20-s − 3·22-s + 6·23-s + 25-s + 4·26-s − 4·28-s + 6·29-s + 2·31-s − 32-s − 3·34-s − 4·35-s − 4·37-s − 5·38-s − 40-s − 3·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.904·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s + 0.223·20-s − 0.639·22-s + 1.25·23-s + 1/5·25-s + 0.784·26-s − 0.755·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s − 0.514·34-s − 0.676·35-s − 0.657·37-s − 0.811·38-s − 0.158·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.054298567\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.054298567\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−9.950585441823304899277092324672, −9.496794239091079548591578063997, −8.891655341825189738159013040687, −7.55386549137356459500397001290, −6.86193526564664030360784937729, −6.12943902760099620494315213595, −5.04061555557317003360241923294, −3.47204628573842727701429029278, −2.64427174967641130853789427610, −0.949027824987604272203929795769,
0.949027824987604272203929795769, 2.64427174967641130853789427610, 3.47204628573842727701429029278, 5.04061555557317003360241923294, 6.12943902760099620494315213595, 6.86193526564664030360784937729, 7.55386549137356459500397001290, 8.891655341825189738159013040687, 9.496794239091079548591578063997, 9.950585441823304899277092324672