| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 2·7-s − 3·8-s − 3·9-s + 9·10-s − 3·11-s − 8·12-s − 3·13-s + 6·14-s + 6·15-s + 3·16-s − 9·17-s + 9·18-s + 5·19-s − 12·20-s + 4·21-s + 9·22-s + 6·24-s − 2·25-s + 9·26-s + 14·27-s − 8·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 0.755·7-s − 1.06·8-s − 9-s + 2.84·10-s − 0.904·11-s − 2.30·12-s − 0.832·13-s + 1.60·14-s + 1.54·15-s + 3/4·16-s − 2.18·17-s + 2.12·18-s + 1.14·19-s − 2.68·20-s + 0.872·21-s + 1.91·22-s + 1.22·24-s − 2/5·25-s + 1.76·26-s + 2.69·27-s − 1.51·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 103 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 95 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 15 T + 173 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 167 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 15 T + 191 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 69 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−13.23610994267001705954558373175, −12.94840009397894493978095356198, −12.11379478170206293955745407270, −11.60864535731955206900512239723, −11.25361490978760122892096508652, −11.08722419550826236265644041303, −10.12933463243780987152001893331, −9.888856913147113375102993549612, −9.102211533828713010939301018281, −8.738425004717429255596542261962, −8.005816357257536141624131095437, −7.80272369775436991917087217809, −6.94244585770451458914106171343, −6.41211906014764542616529081381, −5.49393216015653352470861077787, −4.91231967485079938577327554470, −3.65104599359868815501951116461, −2.59208883639954584899311384770, 0, 0,
2.59208883639954584899311384770, 3.65104599359868815501951116461, 4.91231967485079938577327554470, 5.49393216015653352470861077787, 6.41211906014764542616529081381, 6.94244585770451458914106171343, 7.80272369775436991917087217809, 8.005816357257536141624131095437, 8.738425004717429255596542261962, 9.102211533828713010939301018281, 9.888856913147113375102993549612, 10.12933463243780987152001893331, 11.08722419550826236265644041303, 11.25361490978760122892096508652, 11.60864535731955206900512239723, 12.11379478170206293955745407270, 12.94840009397894493978095356198, 13.23610994267001705954558373175
