| L(s) = 1 | − 2·5-s − 7-s + 4·11-s + 2·13-s + 6·17-s + 4·19-s − 23-s − 25-s − 2·29-s + 8·31-s + 2·35-s − 6·37-s + 6·41-s − 4·43-s − 8·47-s + 49-s + 6·53-s − 8·55-s − 4·59-s + 10·61-s − 4·65-s + 4·67-s − 8·71-s − 6·73-s − 4·77-s + 12·83-s − 12·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 0.520·59-s + 1.28·61-s − 0.496·65-s + 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.455·77-s + 1.31·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.06551032059308, −13.61261879146059, −13.18524673751810, −12.31529067239658, −12.15164238208550, −11.58929885672201, −11.43604137795458, −10.57781075474765, −10.06389567322956, −9.651117851792756, −9.108947964877272, −8.551129819872597, −7.929845325854959, −7.698329479089635, −6.906743484548991, −6.582569020668439, −5.872584954039744, −5.428055740861246, −4.681586423993001, −4.032760444001207, −3.531197255692277, −3.279558041028618, −2.391144198392799, −1.339005790758840, −1.023310109933201, 0,
1.023310109933201, 1.339005790758840, 2.391144198392799, 3.279558041028618, 3.531197255692277, 4.032760444001207, 4.681586423993001, 5.428055740861246, 5.872584954039744, 6.582569020668439, 6.906743484548991, 7.698329479089635, 7.929845325854959, 8.551129819872597, 9.108947964877272, 9.651117851792756, 10.06389567322956, 10.57781075474765, 11.43604137795458, 11.58929885672201, 12.15164238208550, 12.31529067239658, 13.18524673751810, 13.61261879146059, 14.06551032059308
