| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 3·9-s − 2·13-s + 4·15-s + 4·19-s − 4·21-s − 8·23-s + 3·25-s − 4·27-s − 10·29-s − 2·31-s − 4·35-s + 14·37-s + 4·39-s − 12·41-s + 4·43-s − 6·45-s − 6·49-s + 8·53-s − 8·57-s + 18·59-s − 16·61-s + 6·63-s + 4·65-s − 6·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 0.554·13-s + 1.03·15-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 3/5·25-s − 0.769·27-s − 1.85·29-s − 0.359·31-s − 0.676·35-s + 2.30·37-s + 0.640·39-s − 1.87·41-s + 0.609·43-s − 0.894·45-s − 6/7·49-s + 1.09·53-s − 1.05·57-s + 2.34·59-s − 2.04·61-s + 0.755·63-s + 0.496·65-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13838400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13838400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 198 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.141938541201937503561861592057, −7.82351406466869634282047463533, −7.50933929473104756363391763666, −7.49496197242439804236899890035, −6.75756435225147561006332244579, −6.60726340629266522415052807740, −5.95440540124841276377983272944, −5.67746497518047663249582866714, −5.37940507506516526812860901885, −4.97165533591985074678432472638, −4.46254556829077584151830769313, −4.27157908298487974828005476628, −3.67902663584449257347626952077, −3.53045995532553252199680384501, −2.59024059186908339093324335047, −2.30539838373646264061693609889, −1.42507665826195754990578343714, −1.23441897082536577684323572463, 0, 0,
1.23441897082536577684323572463, 1.42507665826195754990578343714, 2.30539838373646264061693609889, 2.59024059186908339093324335047, 3.53045995532553252199680384501, 3.67902663584449257347626952077, 4.27157908298487974828005476628, 4.46254556829077584151830769313, 4.97165533591985074678432472638, 5.37940507506516526812860901885, 5.67746497518047663249582866714, 5.95440540124841276377983272944, 6.60726340629266522415052807740, 6.75756435225147561006332244579, 7.49496197242439804236899890035, 7.50933929473104756363391763666, 7.82351406466869634282047463533, 8.141938541201937503561861592057
