| L(s) = 1 | + 3.02e4·5-s − 2.35e5·7-s − 1.11e7·11-s + 8.04e6·13-s + 1.17e8·17-s + 2.14e8·19-s + 8.30e8·23-s − 3.08e8·25-s + 1.25e9·29-s − 6.15e9·31-s − 7.10e9·35-s − 5.49e9·37-s + 4.67e9·41-s − 7.11e9·43-s − 2.95e10·47-s − 4.16e10·49-s + 2.04e11·53-s − 3.37e11·55-s − 2.99e10·59-s − 1.34e11·61-s + 2.43e11·65-s − 3.48e11·67-s + 1.31e12·71-s − 1.17e12·73-s + 2.62e12·77-s + 1.07e12·79-s + 1.12e12·83-s + ⋯ |
| L(s) = 1 | + 0.864·5-s − 0.755·7-s − 1.90·11-s + 0.462·13-s + 1.18·17-s + 1.04·19-s + 1.16·23-s − 0.252·25-s + 0.390·29-s − 1.24·31-s − 0.653·35-s − 0.352·37-s + 0.153·41-s − 0.171·43-s − 0.399·47-s − 0.429·49-s + 1.26·53-s − 1.64·55-s − 0.0923·59-s − 0.333·61-s + 0.399·65-s − 0.470·67-s + 1.21·71-s − 0.911·73-s + 1.43·77-s + 0.496·79-s + 0.377·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6042 p T + p^{13} T^{2} \) |
| 7 | \( 1 + 33584 p T + p^{13} T^{2} \) |
| 11 | \( 1 + 1016628 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 8049614 T + p^{13} T^{2} \) |
| 17 | \( 1 - 117494622 T + p^{13} T^{2} \) |
| 19 | \( 1 - 214061380 T + p^{13} T^{2} \) |
| 23 | \( 1 - 830555544 T + p^{13} T^{2} \) |
| 29 | \( 1 - 1252400250 T + p^{13} T^{2} \) |
| 31 | \( 1 + 6159350552 T + p^{13} T^{2} \) |
| 37 | \( 1 + 5498191402 T + p^{13} T^{2} \) |
| 41 | \( 1 - 4678687878 T + p^{13} T^{2} \) |
| 43 | \( 1 + 7115013764 T + p^{13} T^{2} \) |
| 47 | \( 1 + 29528776992 T + p^{13} T^{2} \) |
| 53 | \( 1 - 204125042466 T + p^{13} T^{2} \) |
| 59 | \( 1 + 29909821020 T + p^{13} T^{2} \) |
| 61 | \( 1 + 134392006738 T + p^{13} T^{2} \) |
| 67 | \( 1 + 348518801948 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1314335409192 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1178875922326 T + p^{13} T^{2} \) |
| 79 | \( 1 - 1072420659640 T + p^{13} T^{2} \) |
| 83 | \( 1 - 1124025139644 T + p^{13} T^{2} \) |
| 89 | \( 1 + 2235610909530 T + p^{13} T^{2} \) |
| 97 | \( 1 + 14215257165502 T + p^{13} T^{2} \) |
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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
−10.09786316590682164226055130260, −9.382951263980113755599942969543, −8.064762676808942534840313421321, −7.06300396738053541972658154443, −5.71987630735744545464560323771, −5.21117873526440589289883353647, −3.39370313045326164994175883110, −2.59393981979938306762196898919, −1.25785958645119799849380504492, 0,
1.25785958645119799849380504492, 2.59393981979938306762196898919, 3.39370313045326164994175883110, 5.21117873526440589289883353647, 5.71987630735744545464560323771, 7.06300396738053541972658154443, 8.064762676808942534840313421321, 9.382951263980113755599942969543, 10.09786316590682164226055130260
