| L(s) = 1 | + 8·2-s + 18·3-s + 48·4-s + 50·5-s + 144·6-s − 222·7-s + 256·8-s + 243·9-s + 400·10-s − 466·11-s + 864·12-s − 646·13-s − 1.77e3·14-s + 900·15-s + 1.28e3·16-s − 1.52e3·17-s + 1.94e3·18-s + 722·19-s + 2.40e3·20-s − 3.99e3·21-s − 3.72e3·22-s + 524·23-s + 4.60e3·24-s + 1.87e3·25-s − 5.16e3·26-s + 2.91e3·27-s − 1.06e4·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.71·7-s + 1.41·8-s + 9-s + 1.26·10-s − 1.16·11-s + 1.73·12-s − 1.06·13-s − 2.42·14-s + 1.03·15-s + 5/4·16-s − 1.28·17-s + 1.41·18-s + 0.458·19-s + 1.34·20-s − 1.97·21-s − 1.64·22-s + 0.206·23-s + 1.63·24-s + 3/5·25-s − 1.49·26-s + 0.769·27-s − 2.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 222 T + 42460 T^{2} + 222 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 466 T + 242812 T^{2} + 466 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 646 T + 54872 p T^{2} + 646 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1528 T + 3387826 T^{2} + 1528 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 524 T + 6927634 T^{2} - 524 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3930 T + 20012392 T^{2} + 3930 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 14932 T + 108693794 T^{2} + 14932 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11934 T + 106765552 T^{2} + 11934 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5314 T + 154421152 T^{2} + 5314 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6358 T + 221923748 T^{2} + 6358 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8716 T + 477112834 T^{2} + 8716 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 19784 T + 680163994 T^{2} + 19784 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14612 T - 467293490 T^{2} - 14612 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7204 T + 983387442 T^{2} - 7204 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9688 T + 2720503094 T^{2} + 9688 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 102356 T + 6160698982 T^{2} + 102356 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 996 T + 570590614 T^{2} + 996 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 40560 T + 5940669598 T^{2} - 40560 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 111584 T + 9106388314 T^{2} + 111584 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4922 T + 5812039000 T^{2} - 4922 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 48010 T + 10333460064 T^{2} - 48010 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.612235838127843982417445933456, −9.542153772584373043057094966365, −8.858037321957136623811533360043, −8.577431826470078644481238929896, −7.65424469590102643321775695615, −7.37785644560675028506126506285, −6.84271733515001064342263675575, −6.75373518081851922365549390511, −5.81965243137129611871473927180, −5.70834689396731254451277737912, −4.86972979762475077971961375159, −4.79725849120551790494157571681, −3.66681440667551533284133803229, −3.58583782130507667315341738314, −2.84297940525061842822674333770, −2.69538100509700841957075564830, −1.83114335724250469654503182625, −1.75406127235074197456154865999, 0, 0,
1.75406127235074197456154865999, 1.83114335724250469654503182625, 2.69538100509700841957075564830, 2.84297940525061842822674333770, 3.58583782130507667315341738314, 3.66681440667551533284133803229, 4.79725849120551790494157571681, 4.86972979762475077971961375159, 5.70834689396731254451277737912, 5.81965243137129611871473927180, 6.75373518081851922365549390511, 6.84271733515001064342263675575, 7.37785644560675028506126506285, 7.65424469590102643321775695615, 8.577431826470078644481238929896, 8.858037321957136623811533360043, 9.542153772584373043057094966365, 9.612235838127843982417445933456
