| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s + 2·7-s − 4·8-s − 4·10-s − 2·11-s − 6·12-s − 4·14-s − 4·15-s + 5·16-s − 8·17-s + 4·19-s + 6·20-s − 4·21-s + 4·22-s + 6·23-s + 8·24-s − 7·25-s + 2·27-s + 6·28-s − 6·29-s + 8·30-s − 14·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s − 1.26·10-s − 0.603·11-s − 1.73·12-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s + 0.917·19-s + 1.34·20-s − 0.872·21-s + 0.852·22-s + 1.25·23-s + 1.63·24-s − 7/5·25-s + 0.384·27-s + 1.13·28-s − 1.11·29-s + 1.46·30-s − 2.51·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 71 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 119 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 219 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 220 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 186 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.741748965228040116059305076058, −8.726404897655424808948637932301, −8.005652135940026652269571492925, −7.55393352627550083730154100144, −7.20371087313775552158607006375, −7.15097381441409650493219440945, −6.31721371961240335081763044392, −6.21668977498353071511663186502, −5.64529407116854840470095399914, −5.31080603947889639150383716491, −5.19585902846253006247815172391, −4.57609362539724419809850111292, −3.74288402510035051343840566761, −3.48217234130695538968191984774, −2.41752816096609002555097472266, −2.32405619326327404056972640940, −1.71211242000884444400871570650, −1.22664922662716869268755057944, 0, 0,
1.22664922662716869268755057944, 1.71211242000884444400871570650, 2.32405619326327404056972640940, 2.41752816096609002555097472266, 3.48217234130695538968191984774, 3.74288402510035051343840566761, 4.57609362539724419809850111292, 5.19585902846253006247815172391, 5.31080603947889639150383716491, 5.64529407116854840470095399914, 6.21668977498353071511663186502, 6.31721371961240335081763044392, 7.15097381441409650493219440945, 7.20371087313775552158607006375, 7.55393352627550083730154100144, 8.005652135940026652269571492925, 8.726404897655424808948637932301, 8.741748965228040116059305076058
