| L(s) = 1 | + 2.40·2-s + 2.16·3-s + 3.76·4-s − 1.65·5-s + 5.19·6-s − 1.91·7-s + 4.24·8-s + 1.67·9-s − 3.96·10-s + 4.97·11-s + 8.14·12-s + 3.82·13-s − 4.58·14-s − 3.57·15-s + 2.66·16-s + 17-s + 4.02·18-s − 7.52·19-s − 6.22·20-s − 4.13·21-s + 11.9·22-s − 2.57·23-s + 9.18·24-s − 2.27·25-s + 9.18·26-s − 2.86·27-s − 7.20·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.24·3-s + 1.88·4-s − 0.738·5-s + 2.12·6-s − 0.722·7-s + 1.50·8-s + 0.558·9-s − 1.25·10-s + 1.50·11-s + 2.35·12-s + 1.06·13-s − 1.22·14-s − 0.922·15-s + 0.666·16-s + 0.242·17-s + 0.948·18-s − 1.72·19-s − 1.39·20-s − 0.901·21-s + 2.54·22-s − 0.536·23-s + 1.87·24-s − 0.454·25-s + 1.80·26-s − 0.551·27-s − 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.954457480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.954457480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 + 1.91T + 7T^{2} \) |
| 11 | \( 1 - 4.97T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 19 | \( 1 + 7.52T + 19T^{2} \) |
| 23 | \( 1 + 2.57T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 7.93T + 41T^{2} \) |
| 47 | \( 1 - 5.62T + 47T^{2} \) |
| 53 | \( 1 - 1.34T + 53T^{2} \) |
| 59 | \( 1 + 5.27T + 59T^{2} \) |
| 61 | \( 1 + 8.29T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 0.823T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 6.75T + 83T^{2} \) |
| 89 | \( 1 - 7.41T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−10.66969216409541481269557802740, −9.336832543188757645128893064685, −8.627414394655430769344922382990, −7.68616947332110366131066221103, −6.49384001782563197232205238910, −6.12145312539368129493354312841, −4.42248973985780014998811531016, −3.77794214226546053480087449334, −3.29097377191861533397498412395, −2.01210037766059153726528861229,
2.01210037766059153726528861229, 3.29097377191861533397498412395, 3.77794214226546053480087449334, 4.42248973985780014998811531016, 6.12145312539368129493354312841, 6.49384001782563197232205238910, 7.68616947332110366131066221103, 8.627414394655430769344922382990, 9.336832543188757645128893064685, 10.66969216409541481269557802740
