L(s) = 1 | − 3·3-s − 2·5-s + 6·9-s − 5·11-s − 3·13-s + 6·15-s + 17-s + 2·19-s + 8·23-s − 25-s − 9·27-s + 6·29-s − 4·31-s + 15·33-s − 8·37-s + 9·39-s − 6·41-s + 4·43-s − 12·45-s − 10·47-s − 3·51-s − 9·53-s + 10·55-s − 6·57-s − 4·59-s + 4·61-s + 6·65-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.894·5-s + 2·9-s − 1.50·11-s − 0.832·13-s + 1.54·15-s + 0.242·17-s + 0.458·19-s + 1.66·23-s − 1/5·25-s − 1.73·27-s + 1.11·29-s − 0.718·31-s + 2.61·33-s − 1.31·37-s + 1.44·39-s − 0.937·41-s + 0.609·43-s − 1.78·45-s − 1.45·47-s − 0.420·51-s − 1.23·53-s + 1.34·55-s − 0.794·57-s − 0.520·59-s + 0.512·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1856117439\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1856117439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.55686181868722, −13.83903568295580, −13.20326474019582, −12.60492819478058, −12.47132239785690, −11.80230581168329, −11.41058249867521, −10.95005950263045, −10.42892066405044, −10.09812795616875, −9.440272904842737, −8.650899170621421, −7.939499811575638, −7.504864671446105, −7.035393624956508, −6.542370271635661, −5.765041501043960, −5.170447834983602, −4.922159030243845, −4.477403409044873, −3.425884252131551, −2.994840608736967, −1.949318495027014, −1.038163023892590, −0.1978726779821674,
0.1978726779821674, 1.038163023892590, 1.949318495027014, 2.994840608736967, 3.425884252131551, 4.477403409044873, 4.922159030243845, 5.170447834983602, 5.765041501043960, 6.542370271635661, 7.035393624956508, 7.504864671446105, 7.939499811575638, 8.650899170621421, 9.440272904842737, 10.09812795616875, 10.42892066405044, 10.95005950263045, 11.41058249867521, 11.80230581168329, 12.47132239785690, 12.60492819478058, 13.20326474019582, 13.83903568295580, 14.55686181868722