L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s − 2·11-s + 2·12-s + 13-s − 4·16-s − 2·18-s − 19-s + 4·22-s − 2·26-s + 27-s + 4·29-s − 9·31-s + 8·32-s − 2·33-s + 2·36-s − 3·37-s + 2·38-s + 39-s + 10·41-s − 5·43-s − 4·44-s − 6·47-s − 4·48-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 0.603·11-s + 0.577·12-s + 0.277·13-s − 16-s − 0.471·18-s − 0.229·19-s + 0.852·22-s − 0.392·26-s + 0.192·27-s + 0.742·29-s − 1.61·31-s + 1.41·32-s − 0.348·33-s + 1/3·36-s − 0.493·37-s + 0.324·38-s + 0.160·39-s + 1.56·41-s − 0.762·43-s − 0.603·44-s − 0.875·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.330290625897708457512030424135, −7.64806652799736373020768972493, −7.07549960992844820900338549539, −6.20891846751957785484478891111, −5.13864962127656346978799713464, −4.23666237765615592023529733657, −3.18274052552709579425202367333, −2.22020796208357033864499560544, −1.34347248663093181873948752914, 0,
1.34347248663093181873948752914, 2.22020796208357033864499560544, 3.18274052552709579425202367333, 4.23666237765615592023529733657, 5.13864962127656346978799713464, 6.20891846751957785484478891111, 7.07549960992844820900338549539, 7.64806652799736373020768972493, 8.330290625897708457512030424135