| L(s) = 1 | + 5-s − 7-s − 3·9-s − 13-s − 17-s − 4·19-s − 7·23-s − 4·25-s + 3·29-s + 2·31-s − 35-s + 4·37-s − 12·41-s − 11·43-s − 3·45-s + 12·47-s + 49-s − 9·53-s − 3·59-s + 4·61-s + 3·63-s − 65-s + 8·67-s − 5·71-s + 6·73-s + 6·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s − 0.277·13-s − 0.242·17-s − 0.917·19-s − 1.45·23-s − 4/5·25-s + 0.557·29-s + 0.359·31-s − 0.169·35-s + 0.657·37-s − 1.87·41-s − 1.67·43-s − 0.447·45-s + 1.75·47-s + 1/7·49-s − 1.23·53-s − 0.390·59-s + 0.512·61-s + 0.377·63-s − 0.124·65-s + 0.977·67-s − 0.593·71-s + 0.702·73-s + 0.675·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.33604599770048, −12.53272059514306, −12.32872223140340, −11.71895004265629, −11.41471916042944, −10.82376676729619, −10.24027522344746, −9.939171369546122, −9.544814816951209, −8.872076081629041, −8.402820657030061, −8.167886679971554, −7.530975546894926, −6.837920158763922, −6.366996465511130, −6.061932418335824, −5.554048685212493, −4.955400679795199, −4.443542192817129, −3.780057270686354, −3.306672429257240, −2.642360228720667, −2.109434217838748, −1.705318800389623, −0.5992434923404885, 0,
0.5992434923404885, 1.705318800389623, 2.109434217838748, 2.642360228720667, 3.306672429257240, 3.780057270686354, 4.443542192817129, 4.955400679795199, 5.554048685212493, 6.061932418335824, 6.366996465511130, 6.837920158763922, 7.530975546894926, 8.167886679971554, 8.402820657030061, 8.872076081629041, 9.544814816951209, 9.939171369546122, 10.24027522344746, 10.82376676729619, 11.41471916042944, 11.71895004265629, 12.32872223140340, 12.53272059514306, 13.33604599770048
