| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 3·9-s − 6·11-s − 6·13-s + 4·15-s − 8·17-s + 2·19-s + 4·21-s − 8·23-s + 3·25-s + 4·27-s + 2·29-s − 12·31-s − 12·33-s + 4·35-s + 2·37-s − 12·39-s − 14·41-s − 6·43-s + 6·45-s − 8·47-s − 4·49-s − 16·51-s + 12·53-s − 12·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 1.80·11-s − 1.66·13-s + 1.03·15-s − 1.94·17-s + 0.458·19-s + 0.872·21-s − 1.66·23-s + 3/5·25-s + 0.769·27-s + 0.371·29-s − 2.15·31-s − 2.08·33-s + 0.676·35-s + 0.328·37-s − 1.92·39-s − 2.18·41-s − 0.914·43-s + 0.894·45-s − 1.16·47-s − 4/7·49-s − 2.24·51-s + 1.64·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 124 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.103202201267605249542061926087, −7.76177688667222254413923350502, −7.59799929049065973024828667310, −7.20804296654026188462751474235, −6.70058278207679356763305761096, −6.50936628341043363311237784174, −5.92757784137721078706151081570, −5.32764120326536005248220903749, −5.15903143566908425991132953718, −4.91843815644440497652452897532, −4.38349895758822959219982031548, −4.13254558365231013874430078474, −3.26443391341466734443330600764, −3.14208681812751923497089405649, −2.44701737521964160679937682072, −2.25021389140852050145134855535, −1.80967557397962061215883980847, −1.60777651339802500577588911250, 0, 0,
1.60777651339802500577588911250, 1.80967557397962061215883980847, 2.25021389140852050145134855535, 2.44701737521964160679937682072, 3.14208681812751923497089405649, 3.26443391341466734443330600764, 4.13254558365231013874430078474, 4.38349895758822959219982031548, 4.91843815644440497652452897532, 5.15903143566908425991132953718, 5.32764120326536005248220903749, 5.92757784137721078706151081570, 6.50936628341043363311237784174, 6.70058278207679356763305761096, 7.20804296654026188462751474235, 7.59799929049065973024828667310, 7.76177688667222254413923350502, 8.103202201267605249542061926087
