| L(s) = 1 | − 3-s + 4·7-s + 9-s − 13-s − 2·17-s − 8·19-s − 4·21-s + 4·23-s − 27-s + 6·29-s + 8·31-s − 6·37-s + 39-s − 6·41-s − 4·43-s + 4·47-s + 9·49-s + 2·51-s − 6·53-s + 8·57-s − 8·59-s − 10·61-s + 4·63-s + 4·67-s − 4·69-s − 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s − 1.83·19-s − 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s + 1.05·57-s − 1.04·59-s − 1.28·61-s + 0.503·63-s + 0.488·67-s − 0.481·69-s − 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 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 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.39609463636234, −14.89687053372113, −14.35032706006254, −13.71961065017036, −13.36903249324583, −12.44816045958425, −12.21065768785155, −11.63313347950783, −10.96720840594558, −10.70681121555071, −10.23405958268607, −9.412019409272952, −8.655667883605811, −8.344565810189309, −7.815247415460409, −6.972654748885151, −6.557925304922014, −5.962322403252525, −5.055489123995349, −4.695307301625142, −4.382307750193421, −3.356954747131594, −2.430899552878592, −1.808301980662715, −1.072801961514878, 0,
1.072801961514878, 1.808301980662715, 2.430899552878592, 3.356954747131594, 4.382307750193421, 4.695307301625142, 5.055489123995349, 5.962322403252525, 6.557925304922014, 6.972654748885151, 7.815247415460409, 8.344565810189309, 8.655667883605811, 9.412019409272952, 10.23405958268607, 10.70681121555071, 10.96720840594558, 11.63313347950783, 12.21065768785155, 12.44816045958425, 13.36903249324583, 13.71961065017036, 14.35032706006254, 14.89687053372113, 15.39609463636234
