L(s) = 1 | − 2.44·2-s + 3.99·4-s + 2.44·5-s + 2·7-s − 4.89·8-s − 5.99·10-s − 2.44·11-s − 13-s − 4.89·14-s + 3.99·16-s + 7.34·17-s − 19-s + 9.79·20-s + 5.99·22-s + 2.44·23-s + 0.999·25-s + 2.44·26-s + 7.99·28-s + 4.89·29-s − 31-s − 18·34-s + 4.89·35-s + 8·37-s + 2.44·38-s − 11.9·40-s − 4.89·41-s + 11·43-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 1.99·4-s + 1.09·5-s + 0.755·7-s − 1.73·8-s − 1.89·10-s − 0.738·11-s − 0.277·13-s − 1.30·14-s + 0.999·16-s + 1.78·17-s − 0.229·19-s + 2.19·20-s + 1.27·22-s + 0.510·23-s + 0.199·25-s + 0.480·26-s + 1.51·28-s + 0.909·29-s − 0.179·31-s − 3.08·34-s + 0.828·35-s + 1.31·37-s + 0.397·38-s − 1.89·40-s − 0.765·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7295715973\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7295715973\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 7.34T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−11.71630489752962138017706271769, −10.72633054857364640549729516398, −10.01624557525827670252698459237, −9.373503120959489145087639531418, −8.195021880370970742472551747703, −7.60162825894856137206908174096, −6.28792838733343724695047154582, −5.14855857250750191167305147161, −2.65079034130041710876826775177, −1.35764139016953185842768122715,
1.35764139016953185842768122715, 2.65079034130041710876826775177, 5.14855857250750191167305147161, 6.28792838733343724695047154582, 7.60162825894856137206908174096, 8.195021880370970742472551747703, 9.373503120959489145087639531418, 10.01624557525827670252698459237, 10.72633054857364640549729516398, 11.71630489752962138017706271769