L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s + 10-s + 5·11-s − 2·13-s + 3·14-s + 16-s − 3·17-s − 6·19-s − 20-s − 5·22-s + 4·23-s + 25-s + 2·26-s − 3·28-s + 29-s − 3·31-s − 32-s + 3·34-s + 3·35-s − 37-s + 6·38-s + 40-s + 7·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s + 0.316·10-s + 1.50·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s − 1.37·19-s − 0.223·20-s − 1.06·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.566·28-s + 0.185·29-s − 0.538·31-s − 0.176·32-s + 0.514·34-s + 0.507·35-s − 0.164·37-s + 0.973·38-s + 0.158·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8152152977\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8152152977\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.929198720137352528743349049695, −7.936020171375730953473914631474, −6.99879135181147946789051897764, −6.62470444155780626408199418622, −5.95780477626079696380170828616, −4.61551401185348386625175209239, −3.86686308710105270428125307275, −2.98677838409226059495247173192, −1.94000625275420551426618824942, −0.58281538244965167502065518669,
0.58281538244965167502065518669, 1.94000625275420551426618824942, 2.98677838409226059495247173192, 3.86686308710105270428125307275, 4.61551401185348386625175209239, 5.95780477626079696380170828616, 6.62470444155780626408199418622, 6.99879135181147946789051897764, 7.936020171375730953473914631474, 8.929198720137352528743349049695