| L(s) = 1 | − 2.24·2-s − 0.430·3-s + 3.05·4-s − 2.54·5-s + 0.966·6-s − 1.16·7-s − 2.36·8-s − 2.81·9-s + 5.73·10-s − 4.74·11-s − 1.31·12-s + 7.05·13-s + 2.61·14-s + 1.09·15-s − 0.787·16-s + 0.294·17-s + 6.32·18-s − 2.10·19-s − 7.78·20-s + 0.500·21-s + 10.6·22-s + 0.0839·23-s + 1.01·24-s + 1.49·25-s − 15.8·26-s + 2.50·27-s − 3.54·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 0.248·3-s + 1.52·4-s − 1.14·5-s + 0.394·6-s − 0.439·7-s − 0.836·8-s − 0.938·9-s + 1.81·10-s − 1.43·11-s − 0.378·12-s + 1.95·13-s + 0.698·14-s + 0.283·15-s − 0.196·16-s + 0.0714·17-s + 1.49·18-s − 0.482·19-s − 1.73·20-s + 0.109·21-s + 2.27·22-s + 0.0175·23-s + 0.207·24-s + 0.299·25-s − 3.11·26-s + 0.481·27-s − 0.670·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 + 0.430T + 3T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 - 7.05T + 13T^{2} \) |
| 17 | \( 1 - 0.294T + 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 23 | \( 1 - 0.0839T + 23T^{2} \) |
| 29 | \( 1 - 2.00T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.77T + 47T^{2} \) |
| 53 | \( 1 + 1.02T + 53T^{2} \) |
| 59 | \( 1 + 5.69T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 2.67T + 67T^{2} \) |
| 71 | \( 1 - 1.82T + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + 0.146T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.085614101070121564110906589189, −7.79904429656889506335027639734, −6.82199791025890061427939147574, −6.11615132069511910203170085391, −5.28933387399731801398311384150, −4.05620989921748764720618783282, −3.23710217320555778742937309532, −2.27893521762655659826079773520, −0.860816360255064867726808063976, 0,
0.860816360255064867726808063976, 2.27893521762655659826079773520, 3.23710217320555778742937309532, 4.05620989921748764720618783282, 5.28933387399731801398311384150, 6.11615132069511910203170085391, 6.82199791025890061427939147574, 7.79904429656889506335027639734, 8.085614101070121564110906589189
