| L(s) = 1 | − 2·3-s − 2·4-s − 2·5-s − 9-s − 6·11-s + 4·12-s − 6·13-s + 4·15-s + 2·17-s − 12·19-s + 4·20-s + 12·23-s + 3·25-s + 6·27-s − 6·29-s − 12·31-s + 12·33-s + 2·36-s − 4·37-s + 12·39-s + 4·41-s + 4·43-s + 12·44-s + 2·45-s + 6·47-s − 4·51-s + 12·52-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 0.894·5-s − 1/3·9-s − 1.80·11-s + 1.15·12-s − 1.66·13-s + 1.03·15-s + 0.485·17-s − 2.75·19-s + 0.894·20-s + 2.50·23-s + 3/5·25-s + 1.15·27-s − 1.11·29-s − 2.15·31-s + 2.08·33-s + 1/3·36-s − 0.657·37-s + 1.92·39-s + 0.624·41-s + 0.609·43-s + 1.80·44-s + 0.298·45-s + 0.875·47-s − 0.560·51-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 135 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 257 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−11.88162234456419672944908825964, −11.19268451224217980920309784749, −10.84971061629134073981382503730, −10.72899594094127029246533315519, −10.15695282964043647543818608294, −9.274107056346102459764122344395, −8.932596006155112223428001622357, −8.579082003346263727774293825465, −7.81661890654664455796230651952, −7.33830152518063819446024567372, −7.03436080320565253739177080238, −6.08500870730641221226640719789, −5.35566738247323110363826042140, −5.24321247389610054410804711672, −4.61006048249158400129558527219, −4.09947218970825644581307881337, −3.03766218426927581783243183708, −2.34994007735736700716897761349, 0, 0,
2.34994007735736700716897761349, 3.03766218426927581783243183708, 4.09947218970825644581307881337, 4.61006048249158400129558527219, 5.24321247389610054410804711672, 5.35566738247323110363826042140, 6.08500870730641221226640719789, 7.03436080320565253739177080238, 7.33830152518063819446024567372, 7.81661890654664455796230651952, 8.579082003346263727774293825465, 8.932596006155112223428001622357, 9.274107056346102459764122344395, 10.15695282964043647543818608294, 10.72899594094127029246533315519, 10.84971061629134073981382503730, 11.19268451224217980920309784749, 11.88162234456419672944908825964
