L(s) = 1 | + 3-s + 9-s − 6·11-s + 13-s − 3·17-s − 4·19-s + 3·23-s + 27-s + 3·29-s + 5·31-s − 6·33-s − 10·37-s + 39-s − 9·41-s + 43-s − 3·51-s + 9·53-s − 4·57-s + 9·59-s − 11·61-s + 4·67-s + 3·69-s + 12·71-s + 10·73-s + 10·79-s + 81-s + 9·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.277·13-s − 0.727·17-s − 0.917·19-s + 0.625·23-s + 0.192·27-s + 0.557·29-s + 0.898·31-s − 1.04·33-s − 1.64·37-s + 0.160·39-s − 1.40·41-s + 0.152·43-s − 0.420·51-s + 1.23·53-s − 0.529·57-s + 1.17·59-s − 1.40·61-s + 0.488·67-s + 0.361·69-s + 1.42·71-s + 1.17·73-s + 1.12·79-s + 1/9·81-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.66164695584664, −13.85810054270271, −13.57611951221576, −13.24466625038606, −12.58868833532249, −12.25117108194315, −11.50665691880258, −10.80548400317879, −10.52008306007713, −10.11477671740435, −9.428875888736742, −8.676103454822080, −8.470213259170877, −7.958153569436100, −7.331996646761942, −6.698778800916852, −6.337449728421930, −5.297131978207608, −5.082810120395522, −4.411511869954018, −3.628409916621430, −3.099217960982763, −2.310266081986899, −2.065871147385158, −0.8920061836803146, 0,
0.8920061836803146, 2.065871147385158, 2.310266081986899, 3.099217960982763, 3.628409916621430, 4.411511869954018, 5.082810120395522, 5.297131978207608, 6.337449728421930, 6.698778800916852, 7.331996646761942, 7.958153569436100, 8.470213259170877, 8.676103454822080, 9.428875888736742, 10.11477671740435, 10.52008306007713, 10.80548400317879, 11.50665691880258, 12.25117108194315, 12.58868833532249, 13.24466625038606, 13.57611951221576, 13.85810054270271, 14.66164695584664