| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s − 2·10-s − 8·11-s + 6·12-s + 5·13-s − 4·14-s + 2·15-s + 5·16-s + 8·17-s − 6·18-s + 2·19-s + 3·20-s + 4·21-s + 16·22-s + 2·23-s − 8·24-s + 25-s − 10·26-s + 4·27-s + 6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 0.632·10-s − 2.41·11-s + 1.73·12-s + 1.38·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s + 1.94·17-s − 1.41·18-s + 0.458·19-s + 0.670·20-s + 0.872·21-s + 3.41·22-s + 0.417·23-s − 1.63·24-s + 1/5·25-s − 1.96·26-s + 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237970172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237970172\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 162 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 274 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−10.00426136986052802384022504771, −9.953364830239368125571655074213, −9.212867373403171000311331131526, −9.095606485511218683429208318697, −8.405554987148443698619924414055, −8.125110480959164981723335819844, −7.83907588814253103422677825385, −7.71561780347726697298713687871, −7.20344562908946100144309530119, −6.53075073772413365267487489114, −5.91497445398735978294919446427, −5.68124938463567348277028064385, −4.94378442004194954455015025915, −4.67064693646732327554151233917, −3.44595235509003519125432913664, −3.33061698149894243859735840628, −2.62180933416597997887301070516, −2.26126253365041427194897828528, −1.39627103285555046757905291301, −0.917304471445257710921366129008,
0.917304471445257710921366129008, 1.39627103285555046757905291301, 2.26126253365041427194897828528, 2.62180933416597997887301070516, 3.33061698149894243859735840628, 3.44595235509003519125432913664, 4.67064693646732327554151233917, 4.94378442004194954455015025915, 5.68124938463567348277028064385, 5.91497445398735978294919446427, 6.53075073772413365267487489114, 7.20344562908946100144309530119, 7.71561780347726697298713687871, 7.83907588814253103422677825385, 8.125110480959164981723335819844, 8.405554987148443698619924414055, 9.095606485511218683429208318697, 9.212867373403171000311331131526, 9.953364830239368125571655074213, 10.00426136986052802384022504771
