L(s) = 1 | − 2·3-s − 5-s + 7-s + 9-s + 11-s − 2·13-s + 2·15-s − 17-s + 2·19-s − 2·21-s + 4·23-s + 25-s + 4·27-s + 5·29-s + 10·31-s − 2·33-s − 35-s − 2·37-s + 4·39-s + 7·41-s + 6·43-s − 45-s + 7·47-s − 6·49-s + 2·51-s − 5·53-s − 55-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s − 0.242·17-s + 0.458·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.928·29-s + 1.79·31-s − 0.348·33-s − 0.169·35-s − 0.328·37-s + 0.640·39-s + 1.09·41-s + 0.914·43-s − 0.149·45-s + 1.02·47-s − 6/7·49-s + 0.280·51-s − 0.686·53-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 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.42189576513616, −14.15722304191449, −13.64307775083154, −12.75603129233174, −12.46145500089238, −11.96747074678308, −11.51126446624098, −11.14852229739842, −10.59256721188476, −10.16885212093905, −9.438860884746105, −8.947453451460883, −8.299085392425655, −7.755219910455625, −7.205902748918930, −6.588235325489874, −6.179536885533341, −5.548612259760865, −4.855319105286870, −4.617757885997303, −3.971466459613760, −2.919907654654046, −2.643040624490201, −1.381303027282354, −0.8815572046159435, 0,
0.8815572046159435, 1.381303027282354, 2.643040624490201, 2.919907654654046, 3.971466459613760, 4.617757885997303, 4.855319105286870, 5.548612259760865, 6.179536885533341, 6.588235325489874, 7.205902748918930, 7.755219910455625, 8.299085392425655, 8.947453451460883, 9.438860884746105, 10.16885212093905, 10.59256721188476, 11.14852229739842, 11.51126446624098, 11.96747074678308, 12.46145500089238, 12.75603129233174, 13.64307775083154, 14.15722304191449, 14.42189576513616