| L(s) = 1 | − 2·3-s + 3·5-s − 6·7-s + 2·9-s + 11-s − 5·13-s − 6·15-s − 2·17-s + 12·21-s + 7·23-s − 2·25-s − 6·27-s + 5·29-s − 18·31-s − 2·33-s − 18·35-s + 6·37-s + 10·39-s + 18·41-s − 5·43-s + 6·45-s + 47-s + 13·49-s + 4·51-s − 4·53-s + 3·55-s + 2·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s − 2.26·7-s + 2/3·9-s + 0.301·11-s − 1.38·13-s − 1.54·15-s − 0.485·17-s + 2.61·21-s + 1.45·23-s − 2/5·25-s − 1.15·27-s + 0.928·29-s − 3.23·31-s − 0.348·33-s − 3.04·35-s + 0.986·37-s + 1.60·39-s + 2.81·41-s − 0.762·43-s + 0.894·45-s + 0.145·47-s + 13/7·49-s + 0.560·51-s − 0.549·53-s + 0.404·55-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16096144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16096144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| 59 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 31 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 53 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 91 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 83 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 65 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 111 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 167 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 157 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 147 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 137 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 273 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−8.183810068677223725907775167634, −7.74880174501418480416351622482, −7.31566389780050733329977230403, −7.02141969021477093672704730020, −6.57216044732979714779857611362, −6.56064918637373098661216578379, −5.87963832736080334509459610804, −5.82857253152736941345570098221, −5.30079120533470527014894192282, −5.28477365637038007849233194130, −4.44207119154618631833142446474, −4.12707540195388943900960035923, −3.64797509487096491072916419892, −3.11973160474790977534339358271, −2.60958481485575920080146030023, −2.34762787278381892146636415303, −1.70368370476647787999439819349, −1.01168280477197151417615062610, 0, 0,
1.01168280477197151417615062610, 1.70368370476647787999439819349, 2.34762787278381892146636415303, 2.60958481485575920080146030023, 3.11973160474790977534339358271, 3.64797509487096491072916419892, 4.12707540195388943900960035923, 4.44207119154618631833142446474, 5.28477365637038007849233194130, 5.30079120533470527014894192282, 5.82857253152736941345570098221, 5.87963832736080334509459610804, 6.56064918637373098661216578379, 6.57216044732979714779857611362, 7.02141969021477093672704730020, 7.31566389780050733329977230403, 7.74880174501418480416351622482, 8.183810068677223725907775167634
