| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s + 13-s − 2·15-s + 2·17-s − 8·19-s − 4·21-s − 8·23-s − 25-s + 27-s − 2·29-s − 4·31-s + 8·35-s − 10·37-s + 39-s + 2·41-s + 4·43-s − 2·45-s + 12·47-s + 9·49-s + 2·51-s + 6·53-s − 8·57-s − 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 0.485·17-s − 1.83·19-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 + 1.35·35-s − 1.64·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s + 1.75·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 1.05·57-s − 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−10.15504296372293689392693957833, −9.242298798481701557670358922902, −8.449236412144045485016401743930, −7.58162116008273937908413413595, −6.66442414266336138776646532899, −5.77758660498261415913180420481, −4.03149781639917795446220142313, −3.65463435747240401918854167764, −2.29216109830914433710780655355, 0,
2.29216109830914433710780655355, 3.65463435747240401918854167764, 4.03149781639917795446220142313, 5.77758660498261415913180420481, 6.66442414266336138776646532899, 7.58162116008273937908413413595, 8.449236412144045485016401743930, 9.242298798481701557670358922902, 10.15504296372293689392693957833
