| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s − 11-s − 4·13-s − 14-s + 16-s + 4·19-s − 2·20-s − 22-s − 4·23-s − 25-s − 4·26-s − 28-s − 2·29-s − 10·31-s + 32-s + 2·35-s − 6·37-s + 4·38-s − 2·40-s − 4·43-s − 44-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.917·19-s − 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s − 0.784·26-s − 0.188·28-s − 0.371·29-s − 1.79·31-s + 0.176·32-s + 0.338·35-s − 0.986·37-s + 0.648·38-s − 0.316·40-s − 0.609·43-s − 0.150·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.278123263471734382728273256445, −8.085838808436017661666180332862, −7.47160076799445304063607589326, −6.81376239244202158997461277739, −5.64283316816673405468499778451, −4.96360142275870366848623108181, −3.90068439822755419122695958551, −3.23324115647284242077206608800, −2.00597606234881909225076458985, 0,
2.00597606234881909225076458985, 3.23324115647284242077206608800, 3.90068439822755419122695958551, 4.96360142275870366848623108181, 5.64283316816673405468499778451, 6.81376239244202158997461277739, 7.47160076799445304063607589326, 8.085838808436017661666180332862, 9.278123263471734382728273256445
