| L(s) = 1 | − 1.96·2-s − 3-s + 1.87·4-s + 4.10·5-s + 1.96·6-s + 0.241·8-s + 9-s − 8.09·10-s + 4.87·11-s − 1.87·12-s + 2.68·13-s − 4.10·15-s − 4.23·16-s + 3.34·17-s − 1.96·18-s + 0.186·19-s + 7.71·20-s − 9.60·22-s + 5.74·23-s − 0.241·24-s + 11.8·25-s − 5.29·26-s − 27-s + 2.28·29-s + 8.09·30-s − 0.783·31-s + 7.84·32-s + ⋯ |
| L(s) = 1 | − 1.39·2-s − 0.577·3-s + 0.938·4-s + 1.83·5-s + 0.803·6-s + 0.0854·8-s + 0.333·9-s − 2.55·10-s + 1.47·11-s − 0.541·12-s + 0.745·13-s − 1.06·15-s − 1.05·16-s + 0.811·17-s − 0.464·18-s + 0.0426·19-s + 1.72·20-s − 2.04·22-s + 1.19·23-s − 0.0493·24-s + 2.37·25-s − 1.03·26-s − 0.192·27-s + 0.425·29-s + 1.47·30-s − 0.140·31-s + 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.575715783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575715783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 0.186T + 19T^{2} \) |
| 23 | \( 1 - 5.74T + 23T^{2} \) |
| 29 | \( 1 - 2.28T + 29T^{2} \) |
| 31 | \( 1 + 0.783T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 1.36T + 61T^{2} \) |
| 67 | \( 1 + 2.74T + 67T^{2} \) |
| 71 | \( 1 - 6.59T + 71T^{2} \) |
| 73 | \( 1 + 3.81T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.485257278696598302449306269458, −7.19970717004787856814216097848, −6.75261526308314480412230046897, −6.14266232836332169866933108652, −5.45849781868906407221630958406, −4.65661443326395690141412368577, −3.48629314302270333611807659450, −2.28803937645491997692044568999, −1.34468878980685697487639596019, −1.03993478179515962516238115299,
1.03993478179515962516238115299, 1.34468878980685697487639596019, 2.28803937645491997692044568999, 3.48629314302270333611807659450, 4.65661443326395690141412368577, 5.45849781868906407221630958406, 6.14266232836332169866933108652, 6.75261526308314480412230046897, 7.19970717004787856814216097848, 8.485257278696598302449306269458
