| L(s) = 1 | − 2-s + 3-s − 6-s − 8-s − 2·9-s + 6·13-s − 16-s + 9·17-s + 2·18-s − 6·19-s − 9·23-s − 24-s + 3·25-s − 6·26-s − 2·27-s − 13·29-s − 2·31-s + 6·32-s − 9·34-s + 4·37-s + 6·38-s + 6·39-s + 14·41-s − 7·43-s + 9·46-s − 7·47-s − 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 1.66·13-s − 1/4·16-s + 2.18·17-s + 0.471·18-s − 1.37·19-s − 1.87·23-s − 0.204·24-s + 3/5·25-s − 1.17·26-s − 0.384·27-s − 2.41·29-s − 0.359·31-s + 1.06·32-s − 1.54·34-s + 0.657·37-s + 0.973·38-s + 0.960·39-s + 2.18·41-s − 1.06·43-s + 1.32·46-s − 1.02·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 3 p T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 63 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 95 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 77 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 187 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 125 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 53 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 145 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 197 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 3 T - 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 181 T^{2} + p^{2} T^{4} \) |
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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
−7.88553463002950963944431834073, −7.67840089028973141337406431128, −7.67350169139566866917842929542, −6.77322866981739243854373481373, −6.48229684571363421016591261122, −5.95166002701457890369392688363, −5.94692700100053647366424764717, −5.71805591498903709958178308395, −5.06174233819472935436022293022, −4.45882785643029594701420041218, −4.26694779860717531370681081431, −3.65719023396833973202316207326, −3.26595832161314620622222233058, −3.26204536787864239143267894272, −2.47330961011786419743308123253, −2.09548880631912720558403448980, −1.44955327214707115234210094964, −1.20070647018880157829169804729, 0, 0,
1.20070647018880157829169804729, 1.44955327214707115234210094964, 2.09548880631912720558403448980, 2.47330961011786419743308123253, 3.26204536787864239143267894272, 3.26595832161314620622222233058, 3.65719023396833973202316207326, 4.26694779860717531370681081431, 4.45882785643029594701420041218, 5.06174233819472935436022293022, 5.71805591498903709958178308395, 5.94692700100053647366424764717, 5.95166002701457890369392688363, 6.48229684571363421016591261122, 6.77322866981739243854373481373, 7.67350169139566866917842929542, 7.67840089028973141337406431128, 7.88553463002950963944431834073
