| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 13-s + 15-s + 17-s + 4·19-s − 4·21-s − 8·23-s + 25-s + 27-s + 2·29-s − 4·31-s − 4·35-s − 2·37-s − 39-s − 2·41-s − 4·43-s + 45-s + 8·47-s + 9·49-s + 51-s + 10·53-s + 4·57-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.676·35-s − 0.328·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.140·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s + 0.256·61-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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.36289117990370, −12.76069210964805, −12.45020948795138, −11.85650256540539, −11.61121087536471, −10.63354622587290, −10.25900619382173, −9.933266648593893, −9.544936701652473, −9.091510269069622, −8.623079028551626, −8.003345508523808, −7.554341465038455, −6.875970467131732, −6.702395957792867, −6.019039278930806, −5.472682894987783, −5.195625720721995, −4.115324290525412, −3.864922409183006, −3.327211843198321, −2.672096151151796, −2.329838365115035, −1.574494069742026, −0.7795896046992756, 0,
0.7795896046992756, 1.574494069742026, 2.329838365115035, 2.672096151151796, 3.327211843198321, 3.864922409183006, 4.115324290525412, 5.195625720721995, 5.472682894987783, 6.019039278930806, 6.702395957792867, 6.875970467131732, 7.554341465038455, 8.003345508523808, 8.623079028551626, 9.091510269069622, 9.544936701652473, 9.933266648593893, 10.25900619382173, 10.63354622587290, 11.61121087536471, 11.85650256540539, 12.45020948795138, 12.76069210964805, 13.36289117990370
