| L(s) = 1 | − 4·2-s − 6·3-s + 12·4-s + 8·5-s + 24·6-s + 18·7-s − 32·8-s + 27·9-s − 32·10-s + 20·11-s − 72·12-s − 10·13-s − 72·14-s − 48·15-s + 80·16-s + 24·17-s − 108·18-s + 96·20-s − 108·21-s − 80·22-s + 80·23-s + 192·24-s − 82·25-s + 40·26-s − 108·27-s + 216·28-s − 64·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.715·5-s + 1.63·6-s + 0.971·7-s − 1.41·8-s + 9-s − 1.01·10-s + 0.548·11-s − 1.73·12-s − 0.213·13-s − 1.37·14-s − 0.826·15-s + 5/4·16-s + 0.342·17-s − 1.41·18-s + 1.07·20-s − 1.12·21-s − 0.775·22-s + 0.725·23-s + 1.63·24-s − 0.655·25-s + 0.301·26-s − 0.769·27-s + 1.45·28-s − 0.409·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 19 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T + 146 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 18 T + 647 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 20 T + 2642 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 2499 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 80 T + 25814 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 64 T + 49322 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 218 T + 56943 T^{2} + 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 86 T + 55155 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 376 T + 168866 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 254 T + 44463 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 260 T - 7774 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 304 T + 319778 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 540 T + 483538 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 646 T + 500211 T^{2} - 646 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 390 T + 492551 T^{2} + 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 532 T + 647858 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 870 T + 667259 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1762 T + 1756359 T^{2} + 1762 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1824 T + 1958038 T^{2} + 1824 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 60 T + 699358 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1604 T + 2007270 T^{2} + 1604 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.448596014997900896181696538255, −8.417037459864495135404692325071, −7.55282225583898427019191514220, −7.43971123738835964546026159684, −7.00809975861580610488536495799, −6.80796741370118172481118845625, −5.99682947162213409566596285081, −5.96651566724269573354622120197, −5.40891931079639045023926182309, −5.25146701900769884147394475986, −4.47421607053634017586806289058, −4.22190024394487924920300570995, −3.44967812926535985009979042009, −2.99419508588825087753018073578, −2.11937302201226985356775892450, −1.90753615650609810385059617998, −1.21393688494013783180510676488, −1.16734769233290641702361502854, 0, 0,
1.16734769233290641702361502854, 1.21393688494013783180510676488, 1.90753615650609810385059617998, 2.11937302201226985356775892450, 2.99419508588825087753018073578, 3.44967812926535985009979042009, 4.22190024394487924920300570995, 4.47421607053634017586806289058, 5.25146701900769884147394475986, 5.40891931079639045023926182309, 5.96651566724269573354622120197, 5.99682947162213409566596285081, 6.80796741370118172481118845625, 7.00809975861580610488536495799, 7.43971123738835964546026159684, 7.55282225583898427019191514220, 8.417037459864495135404692325071, 8.448596014997900896181696538255
