| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s + 7-s − 4·8-s + 3·9-s + 4·10-s − 2·11-s − 6·12-s + 3·13-s − 2·14-s + 4·15-s + 5·16-s − 2·17-s − 6·18-s − 19-s − 6·20-s − 2·21-s + 4·22-s + 23-s + 8·24-s + 3·25-s − 6·26-s − 4·27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 0.377·7-s − 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s − 1.73·12-s + 0.832·13-s − 0.534·14-s + 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 0.229·19-s − 1.34·20-s − 0.436·21-s + 0.852·22-s + 0.208·23-s + 1.63·24-s + 3/5·25-s − 1.17·26-s − 0.769·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 96 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 138 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 110 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 178 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 228 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−7.86261951477990917339421107870, −7.81288695795899908436022464500, −7.23302495943573180116683692953, −6.93976893976208105947264639126, −6.57210148189662875706516281225, −6.50294989804592786989864098744, −5.76137381635183250970105794642, −5.64862539441396982554805784715, −5.05604947671006150512176127114, −4.80129743167732526968489091511, −4.36385329639549208884865566510, −3.82754167013355105774012465208, −3.38469652423885429698159608264, −3.10589385545724957561097406579, −2.19006797709262293062293465581, −2.08925950855813743833352800638, −1.20512296867367650905062561195, −1.03154509554505000161307815153, 0, 0,
1.03154509554505000161307815153, 1.20512296867367650905062561195, 2.08925950855813743833352800638, 2.19006797709262293062293465581, 3.10589385545724957561097406579, 3.38469652423885429698159608264, 3.82754167013355105774012465208, 4.36385329639549208884865566510, 4.80129743167732526968489091511, 5.05604947671006150512176127114, 5.64862539441396982554805784715, 5.76137381635183250970105794642, 6.50294989804592786989864098744, 6.57210148189662875706516281225, 6.93976893976208105947264639126, 7.23302495943573180116683692953, 7.81288695795899908436022464500, 7.86261951477990917339421107870
