| L(s) = 1 | + 4·7-s + 2·13-s + 2·17-s − 4·19-s − 2·29-s − 10·37-s + 6·41-s − 43-s + 8·47-s + 9·49-s + 2·53-s + 2·61-s − 4·67-s + 8·71-s − 10·73-s + 8·79-s − 16·83-s − 6·89-s + 8·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.371·29-s − 1.64·37-s + 0.937·41-s − 0.152·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.256·61-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s − 1.75·83-s − 0.635·89-s + 0.838·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.72924160200116, −12.95209813007028, −12.66467343373868, −12.03181842610034, −11.61391656555475, −11.19181932400704, −10.65294523552070, −10.42618890151758, −9.768963265740536, −9.007502666504053, −8.718243530544171, −8.270189997884460, −7.765028260867934, −7.279986375110598, −6.801573587071812, −6.000661755322935, −5.682637775761723, −5.049031153966412, −4.617361585187607, −3.978611375037430, −3.599825296266411, −2.690858674094800, −2.131756241470621, −1.521020236403681, −1.015341293046324, 0,
1.015341293046324, 1.521020236403681, 2.131756241470621, 2.690858674094800, 3.599825296266411, 3.978611375037430, 4.617361585187607, 5.049031153966412, 5.682637775761723, 6.000661755322935, 6.801573587071812, 7.279986375110598, 7.765028260867934, 8.270189997884460, 8.718243530544171, 9.007502666504053, 9.768963265740536, 10.42618890151758, 10.65294523552070, 11.19181932400704, 11.61391656555475, 12.03181842610034, 12.66467343373868, 12.95209813007028, 13.72924160200116
