| L(s) = 1 | − 0.899·2-s − 1.19·4-s + 0.803·5-s + 7-s + 2.87·8-s − 0.723·10-s + 2.94·13-s − 0.899·14-s − 0.202·16-s − 3.30·17-s + 7.93·19-s − 0.956·20-s + 7.53·23-s − 4.35·25-s − 2.65·26-s − 1.19·28-s + 0.234·29-s + 3.98·31-s − 5.55·32-s + 2.97·34-s + 0.803·35-s − 0.536·37-s − 7.14·38-s + 2.30·40-s + 4.78·41-s + 3.16·43-s − 6.78·46-s + ⋯ |
| L(s) = 1 | − 0.636·2-s − 0.595·4-s + 0.359·5-s + 0.377·7-s + 1.01·8-s − 0.228·10-s + 0.817·13-s − 0.240·14-s − 0.0506·16-s − 0.801·17-s + 1.82·19-s − 0.213·20-s + 1.57·23-s − 0.870·25-s − 0.520·26-s − 0.224·28-s + 0.0435·29-s + 0.715·31-s − 0.982·32-s + 0.509·34-s + 0.135·35-s − 0.0881·37-s − 1.15·38-s + 0.364·40-s + 0.747·41-s + 0.482·43-s − 0.999·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.604989649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604989649\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.899T + 2T^{2} \) |
| 5 | \( 1 - 0.803T + 5T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 + 3.30T + 17T^{2} \) |
| 19 | \( 1 - 7.93T + 19T^{2} \) |
| 23 | \( 1 - 7.53T + 23T^{2} \) |
| 29 | \( 1 - 0.234T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 + 0.536T + 37T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 + 8.37T + 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 + 8.03T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−7.88192802308302246155771261730, −7.42948991898762414144678213134, −6.58950779450963825244582477832, −5.68543648907089302092145615304, −5.10378529513650249834700205569, −4.37038619418901818659658025582, −3.57150659352279973664892383899, −2.59374763029167183242510206047, −1.44647478656344286619282636696, −0.791654045516962559086576099017,
0.791654045516962559086576099017, 1.44647478656344286619282636696, 2.59374763029167183242510206047, 3.57150659352279973664892383899, 4.37038619418901818659658025582, 5.10378529513650249834700205569, 5.68543648907089302092145615304, 6.58950779450963825244582477832, 7.42948991898762414144678213134, 7.88192802308302246155771261730
