| L(s) = 1 | + 4·7-s − 4·13-s − 8·19-s + 4·23-s + 6·29-s − 8·31-s + 4·37-s − 6·41-s − 4·43-s − 4·47-s + 9·49-s − 12·53-s − 6·61-s − 12·67-s + 16·71-s − 8·79-s − 12·83-s + 10·89-s − 16·91-s − 8·97-s − 14·101-s + 12·103-s + 12·107-s − 10·109-s − 8·113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.10·13-s − 1.83·19-s + 0.834·23-s + 1.11·29-s − 1.43·31-s + 0.657·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s − 1.64·53-s − 0.768·61-s − 1.46·67-s + 1.89·71-s − 0.900·79-s − 1.31·83-s + 1.05·89-s − 1.67·91-s − 0.812·97-s − 1.39·101-s + 1.18·103-s + 1.16·107-s − 0.957·109-s − 0.752·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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
−7.65739347462177858608122825915, −6.93773311332913497977375208770, −6.24263186286545784601861945604, −5.22041361126918224516976910635, −4.77181308518199892712274656054, −4.20456009476023545605210645487, −3.04056135437202175285524365677, −2.13633094575390251570955733800, −1.46694643889267097382892861206, 0,
1.46694643889267097382892861206, 2.13633094575390251570955733800, 3.04056135437202175285524365677, 4.20456009476023545605210645487, 4.77181308518199892712274656054, 5.22041361126918224516976910635, 6.24263186286545784601861945604, 6.93773311332913497977375208770, 7.65739347462177858608122825915
