| L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·11-s − 13-s − 4·17-s − 2·19-s − 4·21-s − 6·23-s − 27-s − 2·29-s + 4·31-s + 2·33-s + 6·37-s + 39-s − 6·41-s + 8·43-s + 8·47-s + 9·49-s + 4·51-s − 10·53-s + 2·57-s + 14·59-s + 10·61-s + 4·63-s + 4·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.970·17-s − 0.458·19-s − 0.872·21-s − 1.25·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.16·47-s + 9/7·49-s + 0.560·51-s − 1.37·53-s + 0.264·57-s + 1.82·59-s + 1.28·61-s + 0.503·63-s + 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.26909986004062, −15.68360115196237, −15.23194991804380, −14.61628375453538, −14.11593144719150, −13.51284205494794, −12.89488699158100, −12.30962864353810, −11.61022564016755, −11.32652100690034, −10.70657931994135, −10.22047469803204, −9.534702478730172, −8.686133455133390, −8.187407710181091, −7.684251668392262, −7.020082634558901, −6.256574386556518, −5.607961984857472, −5.012483181340725, −4.392405480703395, −3.932974821036914, −2.532427670405653, −2.086395457845236, −1.122992970630529, 0,
1.122992970630529, 2.086395457845236, 2.532427670405653, 3.932974821036914, 4.392405480703395, 5.012483181340725, 5.607961984857472, 6.256574386556518, 7.020082634558901, 7.684251668392262, 8.187407710181091, 8.686133455133390, 9.534702478730172, 10.22047469803204, 10.70657931994135, 11.32652100690034, 11.61022564016755, 12.30962864353810, 12.89488699158100, 13.51284205494794, 14.11593144719150, 14.61628375453538, 15.23194991804380, 15.68360115196237, 16.26909986004062
