L(s) = 1 | − 3·3-s − 5-s + 7-s + 6·9-s + 5·11-s + 3·13-s + 3·15-s + 17-s − 6·19-s − 3·21-s + 6·23-s + 25-s − 9·27-s + 9·29-s − 4·31-s − 15·33-s − 35-s + 2·37-s − 9·39-s + 10·43-s − 6·45-s + 47-s + 49-s − 3·51-s − 4·53-s − 5·55-s + 18·57-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s + 0.832·13-s + 0.774·15-s + 0.242·17-s − 1.37·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 0.718·31-s − 2.61·33-s − 0.169·35-s + 0.328·37-s − 1.44·39-s + 1.52·43-s − 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s − 0.674·55-s + 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.76608595924688, −12.63568251270341, −12.14832717272053, −11.58898598942226, −11.40988036392205, −10.87789815660855, −10.57666934563294, −10.16974500310615, −9.374106269285150, −8.776310514192385, −8.741382435174533, −7.774057947598809, −7.341602116446857, −6.803490048332904, −6.340482343216702, −6.066886808440780, −5.606003825731065, −4.746480717230695, −4.500575342151048, −4.161472852571548, −3.449315088487469, −2.780344044935943, −1.738355797465355, −1.243571414013378, −0.8376811141666450, 0,
0.8376811141666450, 1.243571414013378, 1.738355797465355, 2.780344044935943, 3.449315088487469, 4.161472852571548, 4.500575342151048, 4.746480717230695, 5.606003825731065, 6.066886808440780, 6.340482343216702, 6.803490048332904, 7.341602116446857, 7.774057947598809, 8.741382435174533, 8.776310514192385, 9.374106269285150, 10.16974500310615, 10.57666934563294, 10.87789815660855, 11.40988036392205, 11.58898598942226, 12.14832717272053, 12.63568251270341, 12.76608595924688