L(s) = 1 | + 5-s − 5·11-s − 3·17-s − 5·19-s + 6·23-s + 25-s + 10·29-s + 2·31-s + 4·37-s + 3·41-s − 3·43-s + 4·47-s − 7·49-s + 6·53-s − 5·55-s − 3·59-s + 2·61-s + 11·67-s − 14·71-s − 15·73-s − 10·79-s − 12·83-s − 3·85-s − 14·89-s − 5·95-s − 13·97-s + 12·101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.50·11-s − 0.727·17-s − 1.14·19-s + 1.25·23-s + 1/5·25-s + 1.85·29-s + 0.359·31-s + 0.657·37-s + 0.468·41-s − 0.457·43-s + 0.583·47-s − 49-s + 0.824·53-s − 0.674·55-s − 0.390·59-s + 0.256·61-s + 1.34·67-s − 1.66·71-s − 1.75·73-s − 1.12·79-s − 1.31·83-s − 0.325·85-s − 1.48·89-s − 0.512·95-s − 1.31·97-s + 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.67006299247864800041323930423, −6.89932835128991701556296227313, −6.29684097757130235734541369127, −5.49717212119671369995479495529, −4.79281074356321657888352181325, −4.20645732435267191441426447343, −2.74634534584833606286093713898, −2.63781519720586499972838670420, −1.31672849699847440223246128612, 0,
1.31672849699847440223246128612, 2.63781519720586499972838670420, 2.74634534584833606286093713898, 4.20645732435267191441426447343, 4.79281074356321657888352181325, 5.49717212119671369995479495529, 6.29684097757130235734541369127, 6.89932835128991701556296227313, 7.67006299247864800041323930423