| L(s) = 1 | + 2·5-s + 4·7-s − 3·9-s − 2·13-s + 4·19-s + 4·23-s − 25-s − 6·29-s + 4·31-s + 8·35-s + 2·37-s + 6·41-s − 4·43-s − 6·45-s + 9·49-s + 6·53-s + 12·59-s + 10·61-s − 12·63-s − 4·65-s − 4·67-s − 4·71-s + 6·73-s + 12·79-s + 9·81-s + 4·83-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 9-s − 0.554·13-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.35·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.894·45-s + 9/7·49-s + 0.824·53-s + 1.56·59-s + 1.28·61-s − 1.51·63-s − 0.496·65-s − 0.488·67-s − 0.474·71-s + 0.702·73-s + 1.35·79-s + 81-s + 0.439·83-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.663758212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.663758212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.262287466589266701287360658791, −7.71202933945785816476607860166, −6.92397574216590862776428573531, −5.91109494492976391269169882259, −5.32980854506987805197986913493, −4.89803185929025034529624711930, −3.78146246550237005168583590658, −2.64251461061995558898920774055, −2.01683047293254463389404332468, −0.939234788529666549399312646999,
0.939234788529666549399312646999, 2.01683047293254463389404332468, 2.64251461061995558898920774055, 3.78146246550237005168583590658, 4.89803185929025034529624711930, 5.32980854506987805197986913493, 5.91109494492976391269169882259, 6.92397574216590862776428573531, 7.71202933945785816476607860166, 8.262287466589266701287360658791
