L(s) = 1 | + 5-s − 3·7-s + 11-s + 4·19-s + 25-s + 6·29-s + 7·31-s − 3·35-s − 2·37-s − 2·41-s + 43-s + 10·47-s + 2·49-s + 6·53-s + 55-s − 6·59-s − 3·61-s + 7·67-s − 4·71-s − 15·73-s − 3·77-s − 5·79-s + 6·83-s − 10·89-s + 4·95-s + 19·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.301·11-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.25·31-s − 0.507·35-s − 0.328·37-s − 0.312·41-s + 0.152·43-s + 1.45·47-s + 2/7·49-s + 0.824·53-s + 0.134·55-s − 0.781·59-s − 0.384·61-s + 0.855·67-s − 0.474·71-s − 1.75·73-s − 0.341·77-s − 0.562·79-s + 0.658·83-s − 1.05·89-s + 0.410·95-s + 1.92·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−12.83465348901178, −12.29229714412039, −11.99322906316024, −11.59159409696798, −10.94587443871818, −10.32176080322659, −10.10865168489263, −9.741169659547731, −9.114313039815770, −8.844199097004198, −8.336315945676089, −7.613311764519133, −7.262002987294729, −6.671133741247666, −6.328164218538346, −5.871778059534273, −5.401511812413671, −4.728670810783226, −4.323894852190494, −3.611603196954764, −3.141345543337913, −2.710948351687518, −2.154386187173134, −1.260939899069805, −0.8623074149573226, 0,
0.8623074149573226, 1.260939899069805, 2.154386187173134, 2.710948351687518, 3.141345543337913, 3.611603196954764, 4.323894852190494, 4.728670810783226, 5.401511812413671, 5.871778059534273, 6.328164218538346, 6.671133741247666, 7.262002987294729, 7.613311764519133, 8.336315945676089, 8.844199097004198, 9.114313039815770, 9.741169659547731, 10.10865168489263, 10.32176080322659, 10.94587443871818, 11.59159409696798, 11.99322906316024, 12.29229714412039, 12.83465348901178