| L(s) = 1 | − 2-s − 2.23·3-s + 4-s − 3.23·5-s + 2.23·6-s + 0.236·7-s − 8-s + 2.00·9-s + 3.23·10-s + 0.236·11-s − 2.23·12-s + 3.47·13-s − 0.236·14-s + 7.23·15-s + 16-s + 2·17-s − 2.00·18-s + 5.70·19-s − 3.23·20-s − 0.527·21-s − 0.236·22-s − 2.23·23-s + 2.23·24-s + 5.47·25-s − 3.47·26-s + 2.23·27-s + 0.236·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s − 1.44·5-s + 0.912·6-s + 0.0892·7-s − 0.353·8-s + 0.666·9-s + 1.02·10-s + 0.0711·11-s − 0.645·12-s + 0.962·13-s − 0.0630·14-s + 1.86·15-s + 0.250·16-s + 0.485·17-s − 0.471·18-s + 1.30·19-s − 0.723·20-s − 0.115·21-s − 0.0503·22-s − 0.466·23-s + 0.456·24-s + 1.09·25-s − 0.680·26-s + 0.430·27-s + 0.0446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 0.236T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 - 5.23T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.597771190367265092298396861855, −8.670283888555449090702425357769, −7.70819327717467963241719017424, −7.26835824489724665357375500523, −6.10986432752883671598333313405, −5.42287672478591341026314681086, −4.19469688932672413152610489186, −3.26914206818395134575250066423, −1.23307251658556589170753537827, 0,
1.23307251658556589170753537827, 3.26914206818395134575250066423, 4.19469688932672413152610489186, 5.42287672478591341026314681086, 6.10986432752883671598333313405, 7.26835824489724665357375500523, 7.70819327717467963241719017424, 8.670283888555449090702425357769, 9.597771190367265092298396861855
