L(s) = 1 | − 3·7-s − 5·11-s − 13-s + 5·17-s − 2·19-s + 23-s − 10·29-s + 2·31-s + 3·37-s + 9·41-s − 4·43-s − 10·47-s + 2·49-s + 9·53-s − 11·61-s − 4·67-s + 15·71-s − 6·73-s + 15·77-s + 11·79-s − 8·83-s + 11·89-s + 3·91-s + 9·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 1.13·7-s − 1.50·11-s − 0.277·13-s + 1.21·17-s − 0.458·19-s + 0.208·23-s − 1.85·29-s + 0.359·31-s + 0.493·37-s + 1.40·41-s − 0.609·43-s − 1.45·47-s + 2/7·49-s + 1.23·53-s − 1.40·61-s − 0.488·67-s + 1.78·71-s − 0.702·73-s + 1.70·77-s + 1.23·79-s − 0.878·83-s + 1.16·89-s + 0.314·91-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−14.83665536456322, −14.48366318962131, −13.66434312992616, −13.11259030427328, −12.97571614946843, −12.40392037999853, −11.87021942830415, −11.09960255545260, −10.71516416774044, −10.05799295385167, −9.726123189276232, −9.260701911377926, −8.495231897206263, −7.873174782342494, −7.484067757219040, −6.962367715123653, −5.981800632148425, −5.932711224475636, −5.085134449332837, −4.594226880566359, −3.574944055720267, −3.286095347068071, −2.544484630001115, −1.928531858363539, −0.7605072245259315, 0,
0.7605072245259315, 1.928531858363539, 2.544484630001115, 3.286095347068071, 3.574944055720267, 4.594226880566359, 5.085134449332837, 5.932711224475636, 5.981800632148425, 6.962367715123653, 7.484067757219040, 7.873174782342494, 8.495231897206263, 9.260701911377926, 9.726123189276232, 10.05799295385167, 10.71516416774044, 11.09960255545260, 11.87021942830415, 12.40392037999853, 12.97571614946843, 13.11259030427328, 13.66434312992616, 14.48366318962131, 14.83665536456322