| L(s) = 1 | − 3-s − 2·7-s + 9-s + 6·11-s − 2·13-s − 6·17-s − 4·19-s + 2·21-s + 8·23-s − 27-s + 8·31-s − 6·33-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s + 4·47-s − 3·49-s + 6·51-s + 6·53-s + 4·57-s + 6·59-s − 6·61-s − 2·63-s − 8·69-s + 4·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.436·21-s + 1.66·23-s − 0.192·27-s + 1.43·31-s − 1.04·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s − 0.768·61-s − 0.251·63-s − 0.963·69-s + 0.474·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.285817650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.285817650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.065937685945516423160500549979, −8.378538157449977995947227943911, −6.93495664523181794632278927878, −6.74745086677456550394709622316, −6.09536787541905076859475369760, −4.86491973658599309442488184048, −4.26916138881121604221192232437, −3.28122999467983312307906906349, −2.08119807962982229649991921157, −0.74772354250582741256594281217,
0.74772354250582741256594281217, 2.08119807962982229649991921157, 3.28122999467983312307906906349, 4.26916138881121604221192232437, 4.86491973658599309442488184048, 6.09536787541905076859475369760, 6.74745086677456550394709622316, 6.93495664523181794632278927878, 8.378538157449977995947227943911, 9.065937685945516423160500549979
