L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 2·11-s + 16-s + 4·17-s − 6·19-s + 20-s − 2·22-s − 3·23-s + 25-s − 9·29-s − 4·31-s − 32-s − 4·34-s − 4·37-s + 6·38-s − 40-s + 7·41-s − 5·43-s + 2·44-s + 3·46-s − 8·47-s − 50-s + 2·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.603·11-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 1.67·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.657·37-s + 0.973·38-s − 0.158·40-s + 1.09·41-s − 0.762·43-s + 0.301·44-s + 0.442·46-s − 1.16·47-s − 0.141·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.978867865315999140103486121694, −7.43085085363014212731281452820, −6.51215653279629143098878255567, −5.99869648860145192358302637159, −5.18403001049890241512229669344, −4.08226266820595925481841090344, −3.30779312858222897350179871072, −2.13341818101494808635243606638, −1.46450058564780580279055069521, 0,
1.46450058564780580279055069521, 2.13341818101494808635243606638, 3.30779312858222897350179871072, 4.08226266820595925481841090344, 5.18403001049890241512229669344, 5.99869648860145192358302637159, 6.51215653279629143098878255567, 7.43085085363014212731281452820, 7.978867865315999140103486121694