| L(s) = 1 | − 1.66·2-s − 1.15·3-s + 0.770·4-s + 1.93·6-s + 2.43·7-s + 2.04·8-s − 1.65·9-s − 5.75·11-s − 0.893·12-s − 1.59·13-s − 4.05·14-s − 4.94·16-s + 5.98·17-s + 2.75·18-s − 2.82·21-s + 9.57·22-s + 0.940·23-s − 2.37·24-s + 2.65·26-s + 5.39·27-s + 1.87·28-s − 2.61·29-s + 5.26·31-s + 4.14·32-s + 6.67·33-s − 9.96·34-s − 1.27·36-s + ⋯ |
| L(s) = 1 | − 1.17·2-s − 0.669·3-s + 0.385·4-s + 0.788·6-s + 0.920·7-s + 0.723·8-s − 0.551·9-s − 1.73·11-s − 0.258·12-s − 0.442·13-s − 1.08·14-s − 1.23·16-s + 1.45·17-s + 0.649·18-s − 0.616·21-s + 2.04·22-s + 0.196·23-s − 0.484·24-s + 0.520·26-s + 1.03·27-s + 0.354·28-s − 0.486·29-s + 0.946·31-s + 0.732·32-s + 1.16·33-s − 1.70·34-s − 0.212·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4768253244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4768253244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 + 1.15T + 3T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 23 | \( 1 - 0.940T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 + 8.95T + 47T^{2} \) |
| 53 | \( 1 + 2.19T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 1.00T + 67T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−7.86080371179971973051156040140, −7.45328281187911444555217575427, −6.46560191328559063941042043957, −5.50609763972348593591930320635, −5.12226906379260178962265908387, −4.56226506936702915316399510193, −3.22755732048517901531965320170, −2.39842534605707709598620779330, −1.41033245981883767143394211320, −0.43993708233029008854818624015,
0.43993708233029008854818624015, 1.41033245981883767143394211320, 2.39842534605707709598620779330, 3.22755732048517901531965320170, 4.56226506936702915316399510193, 5.12226906379260178962265908387, 5.50609763972348593591930320635, 6.46560191328559063941042043957, 7.45328281187911444555217575427, 7.86080371179971973051156040140
