L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 3·11-s + 6·13-s + 15-s + 4·17-s − 19-s − 21-s − 3·23-s + 25-s − 27-s − 8·29-s + 3·33-s − 35-s + 12·37-s − 6·39-s + 9·43-s − 45-s + 2·47-s − 6·49-s − 4·51-s + 53-s + 3·55-s + 57-s − 6·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.66·13-s + 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.522·33-s − 0.169·35-s + 1.97·37-s − 0.960·39-s + 1.37·43-s − 0.149·45-s + 0.291·47-s − 6/7·49-s − 0.560·51-s + 0.137·53-s + 0.404·55-s + 0.132·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.683098302\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.683098302\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.91178855123616, −12.50038619714108, −11.90264463179851, −11.47277589564674, −11.10727382935304, −10.56679428868984, −10.48810360450955, −9.584006933848156, −9.264190620078041, −8.672670357645560, −7.961551577001215, −7.795395678461657, −7.470454228504245, −6.600002249452454, −6.086313283328998, −5.763105699599839, −5.348744982986570, −4.559289272080714, −4.224217612673676, −3.621797626026672, −3.123206628328498, −2.398793324058172, −1.649336458806602, −1.104475806160263, −0.4147404639288312,
0.4147404639288312, 1.104475806160263, 1.649336458806602, 2.398793324058172, 3.123206628328498, 3.621797626026672, 4.224217612673676, 4.559289272080714, 5.348744982986570, 5.763105699599839, 6.086313283328998, 6.600002249452454, 7.470454228504245, 7.795395678461657, 7.961551577001215, 8.672670357645560, 9.264190620078041, 9.584006933848156, 10.48810360450955, 10.56679428868984, 11.10727382935304, 11.47277589564674, 11.90264463179851, 12.50038619714108, 12.91178855123616