| L(s) = 1 | + 1.64·5-s − 2·7-s + 1.64·11-s + 5.29·13-s − 3.29·17-s − 0.354·19-s − 23-s − 2.29·25-s + 9.29·29-s + 1.29·31-s − 3.29·35-s + 6.93·37-s + 6·41-s − 0.354·43-s − 6·47-s − 3·49-s − 1.64·53-s + 2.70·55-s + 0.354·61-s + 8.70·65-s + 14.9·67-s + 6·71-s − 7.29·73-s − 3.29·77-s − 8.58·79-s + 13.6·83-s − 5.41·85-s + ⋯ |
| L(s) = 1 | + 0.736·5-s − 0.755·7-s + 0.496·11-s + 1.46·13-s − 0.798·17-s − 0.0812·19-s − 0.208·23-s − 0.458·25-s + 1.72·29-s + 0.231·31-s − 0.556·35-s + 1.14·37-s + 0.937·41-s − 0.0540·43-s − 0.875·47-s − 0.428·49-s − 0.226·53-s + 0.365·55-s + 0.0453·61-s + 1.08·65-s + 1.82·67-s + 0.712·71-s − 0.853·73-s − 0.375·77-s − 0.965·79-s + 1.49·83-s − 0.587·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.145091596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.145091596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 + 0.354T + 19T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 0.354T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.354T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 7.29T + 73T^{2} \) |
| 79 | \( 1 + 8.58T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−8.644521081203940132928920917257, −8.078033923840736098921709271125, −6.87244969850281025297828776909, −6.26334574240706933803283295482, −5.95512543701163605323972317764, −4.73954111013446456761199441200, −3.91824892151757401687847009011, −3.03907403530293977978765569322, −2.03313322408017026517600784716, −0.900766288538204409050771170667,
0.900766288538204409050771170667, 2.03313322408017026517600784716, 3.03907403530293977978765569322, 3.91824892151757401687847009011, 4.73954111013446456761199441200, 5.95512543701163605323972317764, 6.26334574240706933803283295482, 6.87244969850281025297828776909, 8.078033923840736098921709271125, 8.644521081203940132928920917257
