| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s − 3·17-s + 18-s + 2·19-s − 21-s − 22-s + 9·23-s + 24-s − 5·25-s + 26-s + 27-s − 28-s − 29-s − 31-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.218·21-s − 0.213·22-s + 1.87·23-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.185·29-s − 0.179·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.440751786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.440751786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.25288730460990, −14.93912965452938, −14.20607961160556, −13.75080919138747, −13.16685218998140, −12.93990329253036, −12.34001108247266, −11.53051160369338, −11.12762262512649, −10.63041299464929, −9.742112735037559, −9.454863186684068, −8.734972426253189, −8.116563994036477, −7.486832824774764, −6.947691874722363, −6.348354399637410, −5.729434290123820, −4.920032418650842, −4.528781590790344, −3.546085845887460, −3.255031711109921, −2.443724123627747, −1.762688327844384, −0.7119841350927628,
0.7119841350927628, 1.762688327844384, 2.443724123627747, 3.255031711109921, 3.546085845887460, 4.528781590790344, 4.920032418650842, 5.729434290123820, 6.348354399637410, 6.947691874722363, 7.486832824774764, 8.116563994036477, 8.734972426253189, 9.454863186684068, 9.742112735037559, 10.63041299464929, 11.12762262512649, 11.53051160369338, 12.34001108247266, 12.93990329253036, 13.16685218998140, 13.75080919138747, 14.20607961160556, 14.93912965452938, 15.25288730460990
