| L(s) = 1 | − 0.381·2-s − 2.61·3-s − 1.85·4-s + 1.23·5-s + 6-s − 4.23·7-s + 1.47·8-s + 3.85·9-s − 0.472·10-s − 3.61·11-s + 4.85·12-s + 1.38·13-s + 1.61·14-s − 3.23·15-s + 3.14·16-s + 6.09·17-s − 1.47·18-s − 1.23·19-s − 2.29·20-s + 11.0·21-s + 1.38·22-s + 4.38·23-s − 3.85·24-s − 3.47·25-s − 0.527·26-s − 2.23·27-s + 7.85·28-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 1.51·3-s − 0.927·4-s + 0.552·5-s + 0.408·6-s − 1.60·7-s + 0.520·8-s + 1.28·9-s − 0.149·10-s − 1.09·11-s + 1.40·12-s + 0.383·13-s + 0.432·14-s − 0.835·15-s + 0.786·16-s + 1.47·17-s − 0.346·18-s − 0.283·19-s − 0.512·20-s + 2.42·21-s + 0.294·22-s + 0.913·23-s − 0.786·24-s − 0.694·25-s − 0.103·26-s − 0.430·27-s + 1.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 - 9.47T + 41T^{2} \) |
| 47 | \( 1 - 1.14T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + 3.85T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 1.85T + 89T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
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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
−9.153977517532590052273936750281, −8.048794747828001243482315470844, −7.12749552882590490191928079526, −6.21033964997663220814003065208, −5.62872462324428161697999258013, −5.10734865966266866981684334966, −3.94741568271863643085453702472, −2.90729373121675889681895518828, −1.03151024896456511366136557881, 0,
1.03151024896456511366136557881, 2.90729373121675889681895518828, 3.94741568271863643085453702472, 5.10734865966266866981684334966, 5.62872462324428161697999258013, 6.21033964997663220814003065208, 7.12749552882590490191928079526, 8.048794747828001243482315470844, 9.153977517532590052273936750281
