| L(s) = 1 | − 8·2-s + 18·3-s + 48·4-s − 4·5-s − 144·6-s − 40·7-s − 256·8-s + 243·9-s + 32·10-s − 160·11-s + 864·12-s − 1.45e3·13-s + 320·14-s − 72·15-s + 1.28e3·16-s − 436·17-s − 1.94e3·18-s − 448·19-s − 192·20-s − 720·21-s + 1.28e3·22-s + 1.05e3·23-s − 4.60e3·24-s − 4.85e3·25-s + 1.16e4·26-s + 2.91e3·27-s − 1.92e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.0715·5-s − 1.63·6-s − 0.308·7-s − 1.41·8-s + 9-s + 0.101·10-s − 0.398·11-s + 1.73·12-s − 2.38·13-s + 0.436·14-s − 0.0826·15-s + 5/4·16-s − 0.365·17-s − 1.41·18-s − 0.284·19-s − 0.107·20-s − 0.356·21-s + 0.563·22-s + 0.417·23-s − 1.63·24-s − 1.55·25-s + 3.37·26-s + 0.769·27-s − 0.462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 4868 T^{2} + 4 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 40 T + 30164 T^{2} + 40 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 160 T + 266902 T^{2} + 160 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 112 p T + 1250394 T^{2} + 112 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 436 T + 80588 T^{2} + 436 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 448 T + 177708 T^{2} + 448 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6364 T + 38900726 T^{2} - 6364 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2360 T + 9315878 T^{2} + 2360 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 524 p T + 207774534 T^{2} + 524 p^{6} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 17476 T + 184431382 T^{2} + 17476 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 23760 T + 395252812 T^{2} + 23760 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 35684 T + 638041978 T^{2} + 35684 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1476 T + 705738564 T^{2} - 1476 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 23092 T + 329788114 T^{2} - 23092 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 70884 T + 2936604790 T^{2} + 70884 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 81824 T + 4336308108 T^{2} + 81824 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15552 T + 698316622 T^{2} - 15552 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 49644 T + 2153290646 T^{2} - 49644 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 102680 T + 8662100564 T^{2} - 102680 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 39096 T + 3615247734 T^{2} - 39096 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21228 T + 7745909020 T^{2} + 21228 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 25772 T + 9703905854 T^{2} + 25772 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.83950399230175841817868486816, −11.65439126184302345771826079200, −10.50807694476052677037821789710, −10.32034059638789620483615028984, −9.669755911709972456676256850501, −9.638787605417820173254910629993, −8.756768626535598038609090765078, −8.437471191599412130403650635700, −7.74427998799202300996848266872, −7.53902721916050615033428893557, −6.66263342669707272181749501124, −6.55025565828904041342278249383, −5.05107157709346456911618752730, −4.81656444009413308506230018046, −3.34026595653235760497990696852, −3.09061740101006725271702968586, −1.97113838551263061840026077997, −1.81804476806768915177807581593, 0, 0,
1.81804476806768915177807581593, 1.97113838551263061840026077997, 3.09061740101006725271702968586, 3.34026595653235760497990696852, 4.81656444009413308506230018046, 5.05107157709346456911618752730, 6.55025565828904041342278249383, 6.66263342669707272181749501124, 7.53902721916050615033428893557, 7.74427998799202300996848266872, 8.437471191599412130403650635700, 8.756768626535598038609090765078, 9.638787605417820173254910629993, 9.669755911709972456676256850501, 10.32034059638789620483615028984, 10.50807694476052677037821789710, 11.65439126184302345771826079200, 11.83950399230175841817868486816
