| L(s) = 1 | − 2-s − 1.36·3-s + 4-s + 1.41·5-s + 1.36·6-s + 1.06·7-s − 8-s − 1.14·9-s − 1.41·10-s + 11-s − 1.36·12-s + 2.62·13-s − 1.06·14-s − 1.92·15-s + 16-s + 5.75·17-s + 1.14·18-s + 7.41·19-s + 1.41·20-s − 1.44·21-s − 22-s + 6.43·23-s + 1.36·24-s − 2.99·25-s − 2.62·26-s + 5.64·27-s + 1.06·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.785·3-s + 0.5·4-s + 0.632·5-s + 0.555·6-s + 0.401·7-s − 0.353·8-s − 0.382·9-s − 0.447·10-s + 0.301·11-s − 0.392·12-s + 0.727·13-s − 0.283·14-s − 0.497·15-s + 0.250·16-s + 1.39·17-s + 0.270·18-s + 1.70·19-s + 0.316·20-s − 0.315·21-s − 0.213·22-s + 1.34·23-s + 0.277·24-s − 0.599·25-s − 0.514·26-s + 1.08·27-s + 0.200·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.410281619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410281619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 - 7.41T + 19T^{2} \) |
| 23 | \( 1 - 6.43T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + 8.28T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 + 0.663T + 47T^{2} \) |
| 53 | \( 1 - 14.5T + 53T^{2} \) |
| 59 | \( 1 - 9.09T + 59T^{2} \) |
| 61 | \( 1 - 1.16T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 5.63T + 73T^{2} \) |
| 79 | \( 1 - 3.53T + 79T^{2} \) |
| 83 | \( 1 + 0.318T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.515020047564677990740113177389, −7.52346469269454907234272260243, −7.09085031842863363190456597217, −5.94133869070468102904935812070, −5.67539988174038226739639623744, −4.99774161871374191543436529156, −3.62606655456825208817695494720, −2.84783422580765549302513484239, −1.53462670266819785004769628970, −0.856176781371131432666943939229,
0.856176781371131432666943939229, 1.53462670266819785004769628970, 2.84783422580765549302513484239, 3.62606655456825208817695494720, 4.99774161871374191543436529156, 5.67539988174038226739639623744, 5.94133869070468102904935812070, 7.09085031842863363190456597217, 7.52346469269454907234272260243, 8.515020047564677990740113177389
