L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 4·11-s − 2·14-s + 16-s + 8·17-s + 6·19-s − 4·22-s + 6·23-s + 2·28-s + 4·29-s − 32-s − 8·34-s − 2·37-s − 6·38-s − 2·41-s + 4·43-s + 4·44-s − 6·46-s − 3·49-s − 10·53-s − 2·56-s − 4·58-s + 4·59-s − 10·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 1.20·11-s − 0.534·14-s + 1/4·16-s + 1.94·17-s + 1.37·19-s − 0.852·22-s + 1.25·23-s + 0.377·28-s + 0.742·29-s − 0.176·32-s − 1.37·34-s − 0.328·37-s − 0.973·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.884·46-s − 3/7·49-s − 1.37·53-s − 0.267·56-s − 0.525·58-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.025585433\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.025585433\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.16949742304532, −13.85678252904631, −12.85010454022209, −12.43709332736075, −11.94561448836546, −11.41186858948533, −11.24251428616080, −10.43525238900811, −9.904652775996557, −9.582219279130410, −8.955506301308104, −8.537715792457992, −7.875731461755266, −7.484208032753178, −7.028602314520817, −6.354055641990871, −5.763971366014735, −5.171480481039022, −4.699201835671582, −3.824627494441918, −3.207166606411550, −2.811239610087525, −1.599228495536421, −1.323380447167499, −0.7265403665557169,
0.7265403665557169, 1.323380447167499, 1.599228495536421, 2.811239610087525, 3.207166606411550, 3.824627494441918, 4.699201835671582, 5.171480481039022, 5.763971366014735, 6.354055641990871, 7.028602314520817, 7.484208032753178, 7.875731461755266, 8.537715792457992, 8.955506301308104, 9.582219279130410, 9.904652775996557, 10.43525238900811, 11.24251428616080, 11.41186858948533, 11.94561448836546, 12.43709332736075, 12.85010454022209, 13.85678252904631, 14.16949742304532