| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 4·11-s − 2·13-s + 2·15-s + 2·17-s − 4·21-s − 25-s − 27-s − 2·29-s − 4·33-s − 8·35-s + 10·37-s + 2·39-s − 6·41-s − 8·43-s − 2·45-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s − 8·55-s − 4·59-s − 14·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 1.35·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 1.07·55-s − 0.520·59-s − 1.79·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−16.53861676036145, −16.23505810864679, −15.20323528231294, −14.87496375917443, −14.61774293357148, −13.78860887294808, −13.19996699976314, −12.26743356831438, −11.80452128059251, −11.63671237363940, −11.06253535819688, −10.37105612785704, −9.671261423732871, −8.975498681721978, −8.252711703295064, −7.743159706856119, −7.279424752641452, −6.480772092288813, −5.776890292134297, −4.967753347802210, −4.478665094869005, −3.930569908013865, −3.028031709132341, −1.779060228872881, −1.239013225900669, 0,
1.239013225900669, 1.779060228872881, 3.028031709132341, 3.930569908013865, 4.478665094869005, 4.967753347802210, 5.776890292134297, 6.480772092288813, 7.279424752641452, 7.743159706856119, 8.252711703295064, 8.975498681721978, 9.671261423732871, 10.37105612785704, 11.06253535819688, 11.63671237363940, 11.80452128059251, 12.26743356831438, 13.19996699976314, 13.78860887294808, 14.61774293357148, 14.87496375917443, 15.20323528231294, 16.23505810864679, 16.53861676036145
