| L(s) = 1 | − 3-s + 1.47·5-s − 7-s + 9-s − 0.875·11-s + 6.74·13-s − 1.47·15-s − 6.71·17-s − 19-s + 21-s + 0.875·23-s − 2.82·25-s − 27-s + 0.599·29-s − 2.87·31-s + 0.875·33-s − 1.47·35-s − 0.294·37-s − 6.74·39-s + 9.98·41-s − 3.41·43-s + 1.47·45-s + 8.34·47-s + 49-s + 6.71·51-s + 12.3·53-s − 1.29·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.659·5-s − 0.377·7-s + 0.333·9-s − 0.263·11-s + 1.86·13-s − 0.380·15-s − 1.62·17-s − 0.229·19-s + 0.218·21-s + 0.182·23-s − 0.564·25-s − 0.192·27-s + 0.111·29-s − 0.516·31-s + 0.152·33-s − 0.249·35-s − 0.0483·37-s − 1.07·39-s + 1.55·41-s − 0.521·43-s + 0.219·45-s + 1.21·47-s + 0.142·49-s + 0.940·51-s + 1.70·53-s − 0.174·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680828489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680828489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + 0.875T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 6.71T + 17T^{2} \) |
| 23 | \( 1 - 0.875T + 23T^{2} \) |
| 29 | \( 1 - 0.599T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 + 0.294T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 - 8.34T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 1.84T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 4.81T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 4.08T + 89T^{2} \) |
| 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
−8.109339474740361213060723602030, −7.06180079166195736813707643815, −6.54536572363945822649982204464, −5.86468062493728284280146209626, −5.46537896807851663572029317781, −4.28653054112453111691406603636, −3.83402133383364309198559209104, −2.62914911744548290906027354813, −1.80298273763158818813662424946, −0.69768603543590329219666474989,
0.69768603543590329219666474989, 1.80298273763158818813662424946, 2.62914911744548290906027354813, 3.83402133383364309198559209104, 4.28653054112453111691406603636, 5.46537896807851663572029317781, 5.86468062493728284280146209626, 6.54536572363945822649982204464, 7.06180079166195736813707643815, 8.109339474740361213060723602030
