| L(s) = 1 | − 4·5-s − 7-s + 2·13-s + 6·17-s + 2·19-s − 2·23-s + 11·25-s + 6·29-s + 4·35-s − 10·37-s + 2·41-s − 10·43-s + 10·47-s + 49-s − 4·53-s + 4·59-s + 10·61-s − 8·65-s − 12·67-s + 10·71-s − 8·73-s + 4·79-s + 12·83-s − 24·85-s − 2·89-s − 2·91-s − 8·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 0.417·23-s + 11/5·25-s + 1.11·29-s + 0.676·35-s − 1.64·37-s + 0.312·41-s − 1.52·43-s + 1.45·47-s + 1/7·49-s − 0.549·53-s + 0.520·59-s + 1.28·61-s − 0.992·65-s − 1.46·67-s + 1.18·71-s − 0.936·73-s + 0.450·79-s + 1.31·83-s − 2.60·85-s − 0.211·89-s − 0.209·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.99299376712250, −13.22523766515511, −12.67546123568260, −12.21620950852012, −11.80442896561229, −11.66795629009277, −10.89442516359667, −10.35161307805214, −10.12822825969642, −9.323524366144948, −8.700100309007184, −8.404130723208905, −7.778014521053650, −7.528994521633192, −6.880410316414651, −6.459514683474224, −5.702409730715024, −5.135306900606736, −4.621585279629182, −3.885969230387221, −3.497628061550424, −3.211408543585519, −2.394331228852631, −1.331034518973831, −0.7904478143742765, 0,
0.7904478143742765, 1.331034518973831, 2.394331228852631, 3.211408543585519, 3.497628061550424, 3.885969230387221, 4.621585279629182, 5.135306900606736, 5.702409730715024, 6.459514683474224, 6.880410316414651, 7.528994521633192, 7.778014521053650, 8.404130723208905, 8.700100309007184, 9.323524366144948, 10.12822825969642, 10.35161307805214, 10.89442516359667, 11.66795629009277, 11.80442896561229, 12.21620950852012, 12.67546123568260, 13.22523766515511, 13.99299376712250
