| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 7-s − 8-s + 9-s + 2·12-s + 4·13-s − 14-s + 16-s − 6·17-s − 18-s + 2·19-s + 2·21-s − 2·24-s − 5·25-s − 4·26-s − 4·27-s + 28-s + 6·29-s − 32-s + 6·34-s + 36-s − 2·37-s − 2·38-s + 8·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s + 0.188·28-s + 1.11·29-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s + 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.27646949907797, −15.96555833947553, −15.29874588802376, −15.00519383820165, −14.16722443463349, −13.73808085467981, −13.34382674384832, −12.61270380365994, −11.72756702240862, −11.30609562196619, −10.78427133865301, −9.994971679718267, −9.457125735265799, −8.787889512312408, −8.509269219312950, −7.916554545480576, −7.384509216287012, −6.457122593942825, −6.091480113501156, −5.003917607240434, −4.242844565030326, −3.435044424964839, −2.842882368899010, −1.980056983457321, −1.391093918828952, 0,
1.391093918828952, 1.980056983457321, 2.842882368899010, 3.435044424964839, 4.242844565030326, 5.003917607240434, 6.091480113501156, 6.457122593942825, 7.384509216287012, 7.916554545480576, 8.509269219312950, 8.787889512312408, 9.457125735265799, 9.994971679718267, 10.78427133865301, 11.30609562196619, 11.72756702240862, 12.61270380365994, 13.34382674384832, 13.73808085467981, 14.16722443463349, 15.00519383820165, 15.29874588802376, 15.96555833947553, 16.27646949907797
