| L(s) = 1 | − 2·3-s + 5-s + 3·7-s + 9-s − 11-s + 6·13-s − 2·15-s − 17-s − 6·19-s − 6·21-s + 4·23-s + 25-s + 4·27-s − 7·29-s + 2·31-s + 2·33-s + 3·35-s + 2·37-s − 12·39-s − 5·41-s − 10·43-s + 45-s − 7·47-s + 2·49-s + 2·51-s − 53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.516·15-s − 0.242·17-s − 1.37·19-s − 1.30·21-s + 0.834·23-s + 1/5·25-s + 0.769·27-s − 1.29·29-s + 0.359·31-s + 0.348·33-s + 0.507·35-s + 0.328·37-s − 1.92·39-s − 0.780·41-s − 1.52·43-s + 0.149·45-s − 1.02·47-s + 2/7·49-s + 0.280·51-s − 0.137·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 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
−14.56877172434105, −14.07579972242764, −13.33962474932535, −13.14103883701580, −12.58835950798092, −11.83206822385828, −11.35406159109695, −11.10058880318679, −10.67257184583763, −10.26363925295376, −9.412348552932890, −8.842349737302879, −8.256950198132076, −8.060109223647732, −7.043648840727792, −6.505360741329216, −6.159522920722717, −5.564643283960987, −4.944992073529202, −4.706272018586318, −3.816449230629881, −3.209772933668180, −2.150001024058252, −1.646389204256270, −0.9492425795983471, 0,
0.9492425795983471, 1.646389204256270, 2.150001024058252, 3.209772933668180, 3.816449230629881, 4.706272018586318, 4.944992073529202, 5.564643283960987, 6.159522920722717, 6.505360741329216, 7.043648840727792, 8.060109223647732, 8.256950198132076, 8.842349737302879, 9.412348552932890, 10.26363925295376, 10.67257184583763, 11.10058880318679, 11.35406159109695, 11.83206822385828, 12.58835950798092, 13.14103883701580, 13.33962474932535, 14.07579972242764, 14.56877172434105
