| L(s) = 1 | + 2·5-s + 2·7-s − 3·9-s + 2·11-s − 2·13-s − 8·17-s − 4·19-s + 3·25-s − 8·29-s − 8·31-s + 4·35-s + 8·37-s + 8·41-s − 6·45-s − 6·47-s + 3·49-s + 4·53-s + 4·55-s + 12·59-s − 4·61-s − 6·63-s − 4·65-s + 4·67-s + 8·71-s − 8·73-s + 4·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 9-s + 0.603·11-s − 0.554·13-s − 1.94·17-s − 0.917·19-s + 3/5·25-s − 1.48·29-s − 1.43·31-s + 0.676·35-s + 1.31·37-s + 1.24·41-s − 0.894·45-s − 0.875·47-s + 3/7·49-s + 0.549·53-s + 0.539·55-s + 1.56·59-s − 0.512·61-s − 0.755·63-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.936·73-s + 0.455·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80281600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80281600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 146 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 171 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 32 T + 447 T^{2} + 32 p T^{3} + p^{2} T^{4} \) |
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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
−7.49335153137804711334519829021, −7.08171640336030040600082398312, −6.85475067549576306891443870152, −6.72055678607305928327271281450, −5.97239832679878174443140089481, −5.92872934779296075144311153274, −5.48470734397030696504631756449, −5.34114956382647680806925129624, −4.81812532869873617353913478964, −4.23795676910371720857407989080, −4.04508350353859279577417416332, −4.03679531562328581872538841621, −2.98013522888253428056400246307, −2.82132077284756603375564499913, −2.28367325112797692524896527361, −2.09962250919695616596709876742, −1.58194872317700331800940517838, −1.13666704448586563345892096132, 0, 0,
1.13666704448586563345892096132, 1.58194872317700331800940517838, 2.09962250919695616596709876742, 2.28367325112797692524896527361, 2.82132077284756603375564499913, 2.98013522888253428056400246307, 4.03679531562328581872538841621, 4.04508350353859279577417416332, 4.23795676910371720857407989080, 4.81812532869873617353913478964, 5.34114956382647680806925129624, 5.48470734397030696504631756449, 5.92872934779296075144311153274, 5.97239832679878174443140089481, 6.72055678607305928327271281450, 6.85475067549576306891443870152, 7.08171640336030040600082398312, 7.49335153137804711334519829021
