| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 3·11-s − 13-s − 15-s − 17-s − 7·19-s + 2·21-s − 4·23-s + 25-s + 27-s + 4·29-s + 11·31-s − 3·33-s − 2·35-s + 9·37-s − 39-s + 4·41-s − 11·43-s − 45-s + 6·47-s − 3·49-s − 51-s − 6·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s − 0.242·17-s − 1.60·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.97·31-s − 0.522·33-s − 0.338·35-s + 1.47·37-s − 0.160·39-s + 0.624·41-s − 1.67·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.140·51-s − 0.824·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−13.16431117673316, −12.83259131948342, −12.38137942215998, −11.76488452645632, −11.46757890442632, −10.80429875680513, −10.48977640810385, −9.956038168734523, −9.576188825615297, −8.804900315267264, −8.355506965201943, −7.995009608242026, −7.893908704322578, −7.081623881861804, −6.487851635428650, −6.197149216165880, −5.357591404443292, −4.831143978712137, −4.260647614516752, −4.194420594411179, −3.184898794793923, −2.656088463685577, −2.273103574048642, −1.604918098906242, −0.7907342163697275, 0,
0.7907342163697275, 1.604918098906242, 2.273103574048642, 2.656088463685577, 3.184898794793923, 4.194420594411179, 4.260647614516752, 4.831143978712137, 5.357591404443292, 6.197149216165880, 6.487851635428650, 7.081623881861804, 7.893908704322578, 7.995009608242026, 8.355506965201943, 8.804900315267264, 9.576188825615297, 9.956038168734523, 10.48977640810385, 10.80429875680513, 11.46757890442632, 11.76488452645632, 12.38137942215998, 12.83259131948342, 13.16431117673316
