| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·11-s − 6·13-s + 5·16-s − 6·17-s − 12·19-s − 8·22-s − 2·23-s − 8·25-s − 12·26-s − 2·29-s − 6·31-s + 6·32-s − 12·34-s − 4·37-s − 24·38-s − 2·43-s − 12·44-s − 4·46-s − 16·50-s − 18·52-s + 14·53-s − 4·58-s − 6·59-s − 12·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1.20·11-s − 1.66·13-s + 5/4·16-s − 1.45·17-s − 2.75·19-s − 1.70·22-s − 0.417·23-s − 8/5·25-s − 2.35·26-s − 0.371·29-s − 1.07·31-s + 1.06·32-s − 2.05·34-s − 0.657·37-s − 3.89·38-s − 0.304·43-s − 1.80·44-s − 0.589·46-s − 2.26·50-s − 2.49·52-s + 1.92·53-s − 0.525·58-s − 0.781·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 119 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 209 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 330 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.500793991931724638653489871972, −8.234808610935603297651295769449, −7.77211430304643258454698573526, −7.46735537077015873032142628226, −6.99050028109064832252254631565, −6.75470517146676555181898002662, −6.13500277211491840435553766669, −6.09815602217500239083051838170, −5.30192695990541561140331364463, −5.20009230671008181719725630340, −4.76506347342434945286987488349, −4.21616604767026033961272550955, −3.99537247150537430840121571645, −3.64510791912941487420056269509, −2.71560876310166327303284802940, −2.45312028885864023011730391452, −2.09079647222061202364621221455, −1.80220743335325392270577860498, 0, 0,
1.80220743335325392270577860498, 2.09079647222061202364621221455, 2.45312028885864023011730391452, 2.71560876310166327303284802940, 3.64510791912941487420056269509, 3.99537247150537430840121571645, 4.21616604767026033961272550955, 4.76506347342434945286987488349, 5.20009230671008181719725630340, 5.30192695990541561140331364463, 6.09815602217500239083051838170, 6.13500277211491840435553766669, 6.75470517146676555181898002662, 6.99050028109064832252254631565, 7.46735537077015873032142628226, 7.77211430304643258454698573526, 8.234808610935603297651295769449, 8.500793991931724638653489871972
