L(s) = 1 | − 3-s − 7-s + 9-s + 5·11-s − 13-s + 7·17-s − 6·19-s + 21-s + 3·23-s − 27-s − 2·29-s − 2·31-s − 5·33-s + 7·37-s + 39-s + 9·41-s + 8·43-s + 10·47-s − 6·49-s − 7·51-s + 5·53-s + 6·57-s − 5·61-s − 63-s + 4·67-s − 3·69-s − 9·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 1.69·17-s − 1.37·19-s + 0.218·21-s + 0.625·23-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.870·33-s + 1.15·37-s + 0.160·39-s + 1.40·41-s + 1.21·43-s + 1.45·47-s − 6/7·49-s − 0.980·51-s + 0.686·53-s + 0.794·57-s − 0.640·61-s − 0.125·63-s + 0.488·67-s − 0.361·69-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.440545468\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.440545468\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−14.34192473916758, −13.87494703085784, −13.01883298723358, −12.73782147186518, −12.19979088942014, −11.87707053317996, −11.17381728986228, −10.81126476155189, −10.21356818262292, −9.650617444954906, −9.138689267109644, −8.855699029369857, −7.841828724856541, −7.544476962525016, −6.886114879138565, −6.250036367733908, −5.977585956629026, −5.365336027411700, −4.557257208193279, −4.061035734706900, −3.579639123782833, −2.764028802437582, −2.012288516494566, −1.124541182145256, −0.6459081257183999,
0.6459081257183999, 1.124541182145256, 2.012288516494566, 2.764028802437582, 3.579639123782833, 4.061035734706900, 4.557257208193279, 5.365336027411700, 5.977585956629026, 6.250036367733908, 6.886114879138565, 7.544476962525016, 7.841828724856541, 8.855699029369857, 9.138689267109644, 9.650617444954906, 10.21356818262292, 10.81126476155189, 11.17381728986228, 11.87707053317996, 12.19979088942014, 12.73782147186518, 13.01883298723358, 13.87494703085784, 14.34192473916758