| L(s) = 1 | − 5-s − 11-s + 5·13-s − 6·17-s − 2·19-s − 3·23-s + 25-s − 9·29-s + 4·31-s + 2·37-s − 9·41-s − 11·43-s + 3·47-s + 3·53-s + 55-s + 8·61-s − 5·65-s − 8·67-s − 6·71-s + 14·73-s + 10·79-s − 12·83-s + 6·85-s − 18·89-s + 2·95-s + 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.38·13-s − 1.45·17-s − 0.458·19-s − 0.625·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s + 0.328·37-s − 1.40·41-s − 1.67·43-s + 0.437·47-s + 0.412·53-s + 0.134·55-s + 1.02·61-s − 0.620·65-s − 0.977·67-s − 0.712·71-s + 1.63·73-s + 1.12·79-s − 1.31·83-s + 0.650·85-s − 1.90·89-s + 0.205·95-s + 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1497164876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1497164876\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.44859572534212, −11.89924649688134, −11.54658214344923, −11.03502824876375, −10.83724319450635, −10.28987585113965, −9.722153617188500, −9.288474692765908, −8.673397870731445, −8.315865188097501, −8.158120622618156, −7.391813560117040, −6.795115238086974, −6.621149334212840, −5.984769536259358, −5.508156784270799, −4.978121826042669, −4.344934249167489, −3.915771181553494, −3.592418187769374, −2.872947237973157, −2.274226474469257, −1.708107414884270, −1.154151601340110, −0.1001481089482836,
0.1001481089482836, 1.154151601340110, 1.708107414884270, 2.274226474469257, 2.872947237973157, 3.592418187769374, 3.915771181553494, 4.344934249167489, 4.978121826042669, 5.508156784270799, 5.984769536259358, 6.621149334212840, 6.795115238086974, 7.391813560117040, 8.158120622618156, 8.315865188097501, 8.673397870731445, 9.288474692765908, 9.722153617188500, 10.28987585113965, 10.83724319450635, 11.03502824876375, 11.54658214344923, 11.89924649688134, 12.44859572534212
