L(s) = 1 | − 5-s − 7-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 25-s + 2·29-s + 8·31-s + 35-s − 2·37-s − 2·41-s + 4·43-s + 49-s + 10·53-s + 4·55-s + 12·59-s + 6·61-s + 2·65-s + 12·67-s − 6·73-s + 4·77-s − 8·79-s − 4·83-s + 2·85-s − 2·89-s + 2·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 1/7·49-s + 1.37·53-s + 0.539·55-s + 1.56·59-s + 0.768·61-s + 0.248·65-s + 1.46·67-s − 0.702·73-s + 0.455·77-s − 0.900·79-s − 0.439·83-s + 0.216·85-s − 0.211·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.217730588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.217730588\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.761403679541310142311328688222, −8.173412186601371566142940384232, −7.35165257679011630657304399485, −6.78643794745841352464669030663, −5.68817332617110801238563692469, −5.00863022908272821636707578096, −4.13106472195914976661110961045, −3.06439471657080577997536290617, −2.34894909423119151865604606934, −0.67621431857077281967426711937,
0.67621431857077281967426711937, 2.34894909423119151865604606934, 3.06439471657080577997536290617, 4.13106472195914976661110961045, 5.00863022908272821636707578096, 5.68817332617110801238563692469, 6.78643794745841352464669030663, 7.35165257679011630657304399485, 8.173412186601371566142940384232, 8.761403679541310142311328688222