| L(s) = 1 | − 3·5-s + 7-s − 7·17-s + 8·19-s − 4·23-s + 4·25-s + 5·29-s − 3·35-s + 3·37-s + 5·41-s + 4·43-s + 49-s − 11·53-s − 4·59-s − 61-s − 4·67-s + 16·71-s + 11·73-s + 4·79-s + 4·83-s + 21·85-s − 10·89-s − 24·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 1.69·17-s + 1.83·19-s − 0.834·23-s + 4/5·25-s + 0.928·29-s − 0.507·35-s + 0.493·37-s + 0.780·41-s + 0.609·43-s + 1/7·49-s − 1.51·53-s − 0.520·59-s − 0.128·61-s − 0.488·67-s + 1.89·71-s + 1.28·73-s + 0.450·79-s + 0.439·83-s + 2.27·85-s − 1.05·89-s − 2.46·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450162557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450162557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.90415182133394, −13.62321287609017, −12.75830979669380, −12.40527469185667, −11.88227416546033, −11.45061751236421, −11.03329116591884, −10.72179034346090, −9.795006595158593, −9.449545987795832, −8.854388089991611, −8.182054581462989, −7.927149223232484, −7.411772346808197, −6.875106313285118, −6.290106833335498, −5.667965726936677, −4.793097283412038, −4.621567394320304, −3.919386483364786, −3.424032907726406, −2.718364513523193, −2.066985894757844, −1.124750816684085, −0.4358237757151782,
0.4358237757151782, 1.124750816684085, 2.066985894757844, 2.718364513523193, 3.424032907726406, 3.919386483364786, 4.621567394320304, 4.793097283412038, 5.667965726936677, 6.290106833335498, 6.875106313285118, 7.411772346808197, 7.927149223232484, 8.182054581462989, 8.854388089991611, 9.449545987795832, 9.795006595158593, 10.72179034346090, 11.03329116591884, 11.45061751236421, 11.88227416546033, 12.40527469185667, 12.75830979669380, 13.62321287609017, 13.90415182133394
