| L(s) = 1 | − 4·5-s + 2·13-s − 8·19-s − 2·23-s + 11·25-s + 8·31-s + 37-s + 2·41-s − 8·47-s + 6·53-s + 10·59-s − 2·61-s − 8·65-s − 12·67-s + 8·71-s − 14·73-s + 8·79-s + 16·83-s − 12·89-s + 32·95-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·115-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.554·13-s − 1.83·19-s − 0.417·23-s + 11/5·25-s + 1.43·31-s + 0.164·37-s + 0.312·41-s − 1.16·47-s + 0.824·53-s + 1.30·59-s − 0.256·61-s − 0.992·65-s − 1.46·67-s + 0.949·71-s − 1.63·73-s + 0.900·79-s + 1.75·83-s − 1.27·89-s + 3.28·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.746·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.63201588331415, −13.19561784140570, −12.61282713095535, −12.30325391956063, −11.71851128254554, −11.39303313077452, −10.96239755629297, −10.34375835194427, −10.09782350125217, −9.116264767587367, −8.734163091468350, −8.233621089756370, −7.977188229367202, −7.426171858031842, −6.700126210116485, −6.476193696726970, −5.803080505536668, −4.964098519209726, −4.482957537530798, −4.029665076858256, −3.672760754930958, −2.924314212659293, −2.376116217817407, −1.461037297919691, −0.6638938126764034, 0,
0.6638938126764034, 1.461037297919691, 2.376116217817407, 2.924314212659293, 3.672760754930958, 4.029665076858256, 4.482957537530798, 4.964098519209726, 5.803080505536668, 6.476193696726970, 6.700126210116485, 7.426171858031842, 7.977188229367202, 8.233621089756370, 8.734163091468350, 9.116264767587367, 10.09782350125217, 10.34375835194427, 10.96239755629297, 11.39303313077452, 11.71851128254554, 12.30325391956063, 12.61282713095535, 13.19561784140570, 13.63201588331415
