L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 11-s + 12-s − 13-s − 16-s + 6·17-s + 18-s − 4·19-s − 22-s + 8·23-s + 3·24-s − 26-s − 27-s − 10·29-s + 5·32-s + 33-s + 6·34-s − 36-s − 6·37-s − 4·38-s + 39-s + 10·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.277·13-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.213·22-s + 1.66·23-s + 0.612·24-s − 0.196·26-s − 0.192·27-s − 1.85·29-s + 0.883·32-s + 0.174·33-s + 1.02·34-s − 1/6·36-s − 0.986·37-s − 0.648·38-s + 0.160·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 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
−16.81045556593852, −16.40755975045793, −15.46376350524279, −15.01981980507851, −14.54305285927140, −14.03205709519323, −13.15375462715251, −12.79558165457163, −12.52533849932577, −11.63347644335024, −11.19844699977157, −10.47438568683167, −9.778984947688950, −9.299058590354278, −8.564270532800161, −7.850014270711203, −7.139516450722529, −6.431797028813529, −5.642860742416598, −5.234625967618866, −4.696128610850488, −3.760984818940426, −3.301398275251033, −2.261846230600368, −1.064159688911846, 0,
1.064159688911846, 2.261846230600368, 3.301398275251033, 3.760984818940426, 4.696128610850488, 5.234625967618866, 5.642860742416598, 6.431797028813529, 7.139516450722529, 7.850014270711203, 8.564270532800161, 9.299058590354278, 9.778984947688950, 10.47438568683167, 11.19844699977157, 11.63347644335024, 12.52533849932577, 12.79558165457163, 13.15375462715251, 14.03205709519323, 14.54305285927140, 15.01981980507851, 15.46376350524279, 16.40755975045793, 16.81045556593852