| L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 13-s + 16-s − 6·17-s − 2·19-s + 22-s + 6·23-s − 5·25-s + 26-s − 9·29-s + 4·31-s + 32-s − 6·34-s + 2·37-s − 2·38-s − 6·41-s − 4·43-s + 44-s + 6·46-s − 6·47-s − 5·50-s + 52-s − 9·58-s − 3·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s + 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.213·22-s + 1.25·23-s − 25-s + 0.196·26-s − 1.67·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.324·38-s − 0.937·41-s − 0.609·43-s + 0.150·44-s + 0.884·46-s − 0.875·47-s − 0.707·50-s + 0.138·52-s − 1.18·58-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.11737313768518753478897799168, −6.60227175063022843294103417050, −6.00666523474469590822246786031, −5.20035681848448274563263099738, −4.53606288756399056878646087116, −3.88869121062273048894330140049, −3.14489936465180768120417808295, −2.22714441386999116729815788889, −1.48346682391232757843772197142, 0,
1.48346682391232757843772197142, 2.22714441386999116729815788889, 3.14489936465180768120417808295, 3.88869121062273048894330140049, 4.53606288756399056878646087116, 5.20035681848448274563263099738, 6.00666523474469590822246786031, 6.60227175063022843294103417050, 7.11737313768518753478897799168
