| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s − 5·11-s + 5·13-s − 16-s + 3·17-s − 19-s + 20-s + 5·22-s − 3·23-s − 4·25-s − 5·26-s + 29-s − 5·32-s − 3·34-s + 3·37-s + 38-s − 3·40-s − 5·41-s − 43-s + 5·44-s + 3·46-s + 4·50-s − 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s − 1.50·11-s + 1.38·13-s − 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 0.625·23-s − 4/5·25-s − 0.980·26-s + 0.185·29-s − 0.883·32-s − 0.514·34-s + 0.493·37-s + 0.162·38-s − 0.474·40-s − 0.780·41-s − 0.152·43-s + 0.753·44-s + 0.442·46-s + 0.565·50-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−8.224084214183849671812381500843, −7.71793366879396218937355031342, −6.81829641441898969929464578299, −5.72779663780015682073103176256, −5.19823273629400202738228518917, −4.13615298318440818427032785089, −3.56338636644165236370575652935, −2.33707184055150419358355728842, −1.12786054204047076608404053599, 0,
1.12786054204047076608404053599, 2.33707184055150419358355728842, 3.56338636644165236370575652935, 4.13615298318440818427032785089, 5.19823273629400202738228518917, 5.72779663780015682073103176256, 6.81829641441898969929464578299, 7.71793366879396218937355031342, 8.224084214183849671812381500843
