| L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·13-s − 8·19-s + 4·21-s + 4·23-s − 27-s + 6·29-s + 8·31-s − 4·37-s − 4·39-s + 6·41-s − 4·43-s − 4·47-s + 9·49-s − 12·53-s + 8·57-s + 6·61-s − 4·63-s − 12·67-s − 4·69-s + 16·71-s + 8·79-s + 81-s + 12·83-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s − 1.83·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s − 1.64·53-s + 1.05·57-s + 0.768·61-s − 0.503·63-s − 1.46·67-s − 0.481·69-s + 1.89·71-s + 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.039137218705779337898366320162, −6.69676559966028498184038513950, −6.58108478534980446814030867766, −6.00471364246196905315013869999, −4.96121140403507523503480637764, −4.15308747711090905962253631775, −3.36210277153781032511787794907, −2.52305561467325089853329107947, −1.15925950305995894267532681030, 0,
1.15925950305995894267532681030, 2.52305561467325089853329107947, 3.36210277153781032511787794907, 4.15308747711090905962253631775, 4.96121140403507523503480637764, 6.00471364246196905315013869999, 6.58108478534980446814030867766, 6.69676559966028498184038513950, 8.039137218705779337898366320162
