| L(s) = 1 | − 2.06·2-s − 3-s + 2.27·4-s − 0.238·5-s + 2.06·6-s + 7-s − 0.558·8-s + 9-s + 0.491·10-s − 5.40·11-s − 2.27·12-s + 0.238·13-s − 2.06·14-s + 0.238·15-s − 3.38·16-s − 17-s − 2.06·18-s + 7.89·19-s − 0.540·20-s − 21-s + 11.1·22-s + 6.38·23-s + 0.558·24-s − 4.94·25-s − 0.491·26-s − 27-s + 2.27·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 0.577·3-s + 1.13·4-s − 0.106·5-s + 0.843·6-s + 0.377·7-s − 0.197·8-s + 0.333·9-s + 0.155·10-s − 1.62·11-s − 0.655·12-s + 0.0660·13-s − 0.552·14-s + 0.0614·15-s − 0.846·16-s − 0.242·17-s − 0.487·18-s + 1.81·19-s − 0.120·20-s − 0.218·21-s + 2.38·22-s + 1.33·23-s + 0.113·24-s − 0.988·25-s − 0.0964·26-s − 0.192·27-s + 0.429·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4990363148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4990363148\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 5 | \( 1 + 0.238T + 5T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 - 0.238T + 13T^{2} \) |
| 19 | \( 1 - 7.89T + 19T^{2} \) |
| 23 | \( 1 - 6.38T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 6.18T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 + 6.30T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 6.18T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 - 5.45T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−11.18183079890896129994370095848, −10.39776777972413569729574069695, −9.738720151306175736757794837053, −8.660182292476093748064227815120, −7.73047959398978833843427895427, −7.18032724620857925640044392391, −5.68641473272541393924609196502, −4.67942543603881613674176278989, −2.65319336574567211412948083003, −0.903628635718627513723344422918,
0.903628635718627513723344422918, 2.65319336574567211412948083003, 4.67942543603881613674176278989, 5.68641473272541393924609196502, 7.18032724620857925640044392391, 7.73047959398978833843427895427, 8.660182292476093748064227815120, 9.738720151306175736757794837053, 10.39776777972413569729574069695, 11.18183079890896129994370095848
