| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 7-s − 3·8-s + 9-s − 2·11-s + 12-s + 5·13-s + 14-s − 16-s − 4·17-s + 18-s − 21-s − 2·22-s − 4·23-s + 3·24-s − 5·25-s + 5·26-s − 27-s − 28-s − 8·29-s − 3·31-s + 5·32-s + 2·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.612·24-s − 25-s + 0.980·26-s − 0.192·27-s − 0.188·28-s − 1.48·29-s − 0.538·31-s + 0.883·32-s + 0.348·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.490859406317181910865214053483, −8.595490268432358233091447782559, −7.88766191356537491665576331142, −6.62079287213053781803056519115, −5.84047896161452237997455144358, −5.18046014163099724265079503817, −4.20637327455860491087854991343, −3.48638822023419989452758004968, −1.86277933111965985175848882827, 0,
1.86277933111965985175848882827, 3.48638822023419989452758004968, 4.20637327455860491087854991343, 5.18046014163099724265079503817, 5.84047896161452237997455144358, 6.62079287213053781803056519115, 7.88766191356537491665576331142, 8.595490268432358233091447782559, 9.490859406317181910865214053483
