L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 4·11-s + 6·13-s − 15-s + 6·17-s − 6·19-s + 21-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·33-s − 35-s + 8·37-s − 6·39-s − 10·41-s + 4·43-s + 45-s − 12·47-s + 49-s − 6·51-s + 6·53-s − 4·55-s + 6·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.258·15-s + 1.45·17-s − 1.37·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.169·35-s + 1.31·37-s − 0.960·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.787373049\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.787373049\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.89604943104794, −12.68404987010661, −12.11604208005801, −11.30505878498678, −11.21136651621640, −10.51712432214801, −10.35731760801057, −9.738331669667931, −9.362473079802207, −8.583842839261194, −8.300277281357116, −7.810508269960660, −7.179259416857433, −6.636794831531119, −6.161190107946040, −5.704974527815846, −5.409852010932724, −4.807986018979180, −4.114214474866144, −3.489878919162231, −3.221815012674717, −2.266384019961750, −1.843904451524048, −1.071439387405652, −0.4240072273273520,
0.4240072273273520, 1.071439387405652, 1.843904451524048, 2.266384019961750, 3.221815012674717, 3.489878919162231, 4.114214474866144, 4.807986018979180, 5.409852010932724, 5.704974527815846, 6.161190107946040, 6.636794831531119, 7.179259416857433, 7.810508269960660, 8.300277281357116, 8.583842839261194, 9.362473079802207, 9.738331669667931, 10.35731760801057, 10.51712432214801, 11.21136651621640, 11.30505878498678, 12.11604208005801, 12.68404987010661, 12.89604943104794