L(s) = 1 | − 0.911·2-s − 0.648·3-s − 1.16·4-s − 1.23·5-s + 0.590·6-s + 1.76·7-s + 2.88·8-s − 2.57·9-s + 1.12·10-s + 0.758·12-s + 4.35·13-s − 1.60·14-s + 0.801·15-s − 0.292·16-s − 7.66·17-s + 2.35·18-s − 4.10·19-s + 1.44·20-s − 1.14·21-s + 8.58·23-s − 1.87·24-s − 3.47·25-s − 3.96·26-s + 3.61·27-s − 2.05·28-s − 8.70·29-s − 0.730·30-s + ⋯ |
L(s) = 1 | − 0.644·2-s − 0.374·3-s − 0.584·4-s − 0.552·5-s + 0.241·6-s + 0.665·7-s + 1.02·8-s − 0.859·9-s + 0.356·10-s + 0.218·12-s + 1.20·13-s − 0.428·14-s + 0.206·15-s − 0.0731·16-s − 1.85·17-s + 0.554·18-s − 0.942·19-s + 0.323·20-s − 0.249·21-s + 1.78·23-s − 0.382·24-s − 0.694·25-s − 0.778·26-s + 0.696·27-s − 0.389·28-s − 1.61·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6218394612\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6218394612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + 0.911T + 2T^{2} \) |
| 3 | \( 1 + 0.648T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 1.76T + 7T^{2} \) |
| 13 | \( 1 - 4.35T + 13T^{2} \) |
| 17 | \( 1 + 7.66T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 0.393T + 31T^{2} \) |
| 37 | \( 1 - 0.904T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 - 9.86T + 43T^{2} \) |
| 47 | \( 1 - 2.45T + 47T^{2} \) |
| 53 | \( 1 - 1.96T + 53T^{2} \) |
| 59 | \( 1 - 0.899T + 59T^{2} \) |
| 61 | \( 1 - 8.22T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 + 1.25T + 83T^{2} \) |
| 89 | \( 1 - 9.43T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.252737889059619411707124247330, −8.834703676214825080767660527002, −8.245092071590808783261756552187, −7.38812061893960787432242396573, −6.36867531408990992887557808190, −5.38612151035955295609586984612, −4.48368278288078183158438561241, −3.73802721723972432748080169462, −2.12592001731412998408756567119, −0.64423993331236591542428735115,
0.64423993331236591542428735115, 2.12592001731412998408756567119, 3.73802721723972432748080169462, 4.48368278288078183158438561241, 5.38612151035955295609586984612, 6.36867531408990992887557808190, 7.38812061893960787432242396573, 8.245092071590808783261756552187, 8.834703676214825080767660527002, 9.252737889059619411707124247330