| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s − 11-s + 2·14-s + 16-s − 8·19-s − 20-s − 22-s − 4·23-s + 25-s + 2·28-s − 6·29-s + 4·31-s + 32-s − 2·35-s + 2·37-s − 8·38-s − 40-s + 10·43-s − 44-s − 4·46-s − 8·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.534·14-s + 1/4·16-s − 1.83·19-s − 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.338·35-s + 0.328·37-s − 1.29·38-s − 0.158·40-s + 1.52·43-s − 0.150·44-s − 0.589·46-s − 1.16·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.06134058555716, −12.46231923676704, −12.11928523931774, −11.62999503800507, −11.04356209992348, −10.94773636138623, −10.33093809770855, −9.896570457424939, −9.183740595642253, −8.724986702819676, −8.133767982717406, −7.879126322266143, −7.417615881332638, −6.772491789110775, −6.275728768112304, −5.886250455072637, −5.307069177253293, −4.671424335802391, −4.393166269057756, −3.914212484453402, −3.354061992855091, −2.643728057988138, −2.084777821932219, −1.718603053862008, −0.7830695120786520, 0,
0.7830695120786520, 1.718603053862008, 2.084777821932219, 2.643728057988138, 3.354061992855091, 3.914212484453402, 4.393166269057756, 4.671424335802391, 5.307069177253293, 5.886250455072637, 6.275728768112304, 6.772491789110775, 7.417615881332638, 7.879126322266143, 8.133767982717406, 8.724986702819676, 9.183740595642253, 9.896570457424939, 10.33093809770855, 10.94773636138623, 11.04356209992348, 11.62999503800507, 12.11928523931774, 12.46231923676704, 13.06134058555716
