L(s) = 1 | − 1.47·2-s + 0.718·3-s + 0.162·4-s + 5-s − 1.05·6-s + 2.70·8-s − 2.48·9-s − 1.47·10-s − 11-s + 0.116·12-s − 1.35·13-s + 0.718·15-s − 4.29·16-s − 0.542·17-s + 3.65·18-s + 0.199·19-s + 0.162·20-s + 1.47·22-s − 3.97·23-s + 1.94·24-s + 25-s + 1.98·26-s − 3.94·27-s + 2.25·29-s − 1.05·30-s + 6.03·31-s + 0.916·32-s + ⋯ |
L(s) = 1 | − 1.03·2-s + 0.415·3-s + 0.0811·4-s + 0.447·5-s − 0.431·6-s + 0.955·8-s − 0.827·9-s − 0.465·10-s − 0.301·11-s + 0.0336·12-s − 0.375·13-s + 0.185·15-s − 1.07·16-s − 0.131·17-s + 0.860·18-s + 0.0457·19-s + 0.0363·20-s + 0.313·22-s − 0.828·23-s + 0.396·24-s + 0.200·25-s + 0.389·26-s − 0.758·27-s + 0.418·29-s − 0.193·30-s + 1.08·31-s + 0.161·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9396709978\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9396709978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 3 | \( 1 - 0.718T + 3T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 0.542T + 17T^{2} \) |
| 19 | \( 1 - 0.199T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 - 4.07T + 41T^{2} \) |
| 43 | \( 1 + 9.85T + 43T^{2} \) |
| 47 | \( 1 - 7.86T + 47T^{2} \) |
| 53 | \( 1 - 9.02T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 - 6.96T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 0.363T + 79T^{2} \) |
| 83 | \( 1 - 3.87T + 83T^{2} \) |
| 89 | \( 1 + 3.98T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.821331669135820766694782708521, −8.198011326440404728719990504106, −7.70860164033060419711879498163, −6.72080659248026085498703358105, −5.83959561137455883157890285791, −4.95901636009357216000117659816, −4.04933340045532889902479248654, −2.80135659613892074754603733760, −2.04203097334280111109037083130, −0.68452516381464898816598356508,
0.68452516381464898816598356508, 2.04203097334280111109037083130, 2.80135659613892074754603733760, 4.04933340045532889902479248654, 4.95901636009357216000117659816, 5.83959561137455883157890285791, 6.72080659248026085498703358105, 7.70860164033060419711879498163, 8.198011326440404728719990504106, 8.821331669135820766694782708521