| L(s) = 1 | − 256·2-s − 6.56e3·3-s + 6.55e4·4-s + 6.45e5·5-s + 1.67e6·6-s + 3.97e6·7-s − 1.67e7·8-s + 4.30e7·9-s − 1.65e8·10-s − 5.00e8·11-s − 4.29e8·12-s − 5.42e9·13-s − 1.01e9·14-s − 4.23e9·15-s + 4.29e9·16-s − 5.46e9·17-s − 1.10e10·18-s − 5.38e10·19-s + 4.22e10·20-s − 2.60e10·21-s + 1.28e11·22-s + 5.78e11·23-s + 1.10e11·24-s − 3.46e11·25-s + 1.38e12·26-s − 2.82e11·27-s + 2.60e11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.738·5-s + 0.408·6-s + 0.260·7-s − 0.353·8-s + 1/3·9-s − 0.522·10-s − 0.703·11-s − 0.288·12-s − 1.84·13-s − 0.184·14-s − 0.426·15-s + 1/4·16-s − 0.190·17-s − 0.235·18-s − 0.727·19-s + 0.369·20-s − 0.150·21-s + 0.497·22-s + 1.54·23-s + 0.204·24-s − 0.454·25-s + 1.30·26-s − 0.192·27-s + 0.130·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{8} T \) |
| 3 | \( 1 + p^{8} T \) |
| good | 5 | \( 1 - 25806 p^{2} T + p^{17} T^{2} \) |
| 7 | \( 1 - 567776 p T + p^{17} T^{2} \) |
| 11 | \( 1 + 45460788 p T + p^{17} T^{2} \) |
| 13 | \( 1 + 5425661314 T + p^{17} T^{2} \) |
| 17 | \( 1 + 5466992958 T + p^{17} T^{2} \) |
| 19 | \( 1 + 53889877060 T + p^{17} T^{2} \) |
| 23 | \( 1 - 578906836536 T + p^{17} T^{2} \) |
| 29 | \( 1 + 4619583681690 T + p^{17} T^{2} \) |
| 31 | \( 1 + 6802815567448 T + p^{17} T^{2} \) |
| 37 | \( 1 + 19571909422138 T + p^{17} T^{2} \) |
| 41 | \( 1 - 57213620756922 T + p^{17} T^{2} \) |
| 43 | \( 1 + 24501250225084 T + p^{17} T^{2} \) |
| 47 | \( 1 - 184283998832832 T + p^{17} T^{2} \) |
| 53 | \( 1 + 206542562280354 T + p^{17} T^{2} \) |
| 59 | \( 1 + 418648048246140 T + p^{17} T^{2} \) |
| 61 | \( 1 - 2501287878088382 T + p^{17} T^{2} \) |
| 67 | \( 1 + 145692866050948 T + p^{17} T^{2} \) |
| 71 | \( 1 + 5364313152664248 T + p^{17} T^{2} \) |
| 73 | \( 1 - 3302058927938186 T + p^{17} T^{2} \) |
| 79 | \( 1 - 22067463278260760 T + p^{17} T^{2} \) |
| 83 | \( 1 - 20438378406354876 T + p^{17} T^{2} \) |
| 89 | \( 1 + 56063805950152710 T + p^{17} T^{2} \) |
| 97 | \( 1 + 118254406396110718 T + p^{17} T^{2} \) |
| 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
−17.68693402654015723782842418374, −16.81530778009832749235234374319, −14.92857252611761500445607255585, −12.75190728140454758704120009201, −10.90766785822369459716757028814, −9.483467487472704116543893836061, −7.28870293225689242567611511459, −5.32472584341642081578733119505, −2.11022976097817694853914365166, 0,
2.11022976097817694853914365166, 5.32472584341642081578733119505, 7.28870293225689242567611511459, 9.483467487472704116543893836061, 10.90766785822369459716757028814, 12.75190728140454758704120009201, 14.92857252611761500445607255585, 16.81530778009832749235234374319, 17.68693402654015723782842418374
