| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 3·11-s − 13-s + 4·14-s + 16-s + 2·19-s − 3·22-s + 9·23-s + 26-s − 4·28-s − 6·29-s − 10·31-s − 32-s − 7·37-s − 2·38-s + 6·41-s − 10·43-s + 3·44-s − 9·46-s − 9·47-s + 9·49-s − 52-s − 6·53-s + 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 0.904·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.458·19-s − 0.639·22-s + 1.87·23-s + 0.196·26-s − 0.755·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s − 1.15·37-s − 0.324·38-s + 0.937·41-s − 1.52·43-s + 0.452·44-s − 1.32·46-s − 1.31·47-s + 9/7·49-s − 0.138·52-s − 0.824·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−9.356110654915629502850291910668, −8.683828868126184452501008466835, −7.37722472335971541888970783096, −6.92315963083335083800486379426, −6.12944776044565246412216766706, −5.12775020532760155184495038376, −3.64777311545877993789541794244, −3.04210092197633002590953011250, −1.55926991803029948080395915743, 0,
1.55926991803029948080395915743, 3.04210092197633002590953011250, 3.64777311545877993789541794244, 5.12775020532760155184495038376, 6.12944776044565246412216766706, 6.92315963083335083800486379426, 7.37722472335971541888970783096, 8.683828868126184452501008466835, 9.356110654915629502850291910668
