| L(s) = 1 | − 2·3-s + 4·7-s + 2·9-s − 2·11-s + 4·17-s − 2·19-s − 8·21-s + 2·23-s − 6·27-s − 8·29-s + 8·31-s + 4·33-s + 18·37-s + 6·41-s − 14·43-s − 10·47-s + 3·49-s − 8·51-s − 2·53-s + 4·57-s − 2·59-s − 10·61-s + 8·63-s − 8·67-s − 4·69-s − 20·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 2/3·9-s − 0.603·11-s + 0.970·17-s − 0.458·19-s − 1.74·21-s + 0.417·23-s − 1.15·27-s − 1.48·29-s + 1.43·31-s + 0.696·33-s + 2.95·37-s + 0.937·41-s − 2.13·43-s − 1.45·47-s + 3/7·49-s − 1.12·51-s − 0.274·53-s + 0.529·57-s − 0.260·59-s − 1.28·61-s + 1.00·63-s − 0.977·67-s − 0.481·69-s − 2.37·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 18 T + 150 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 151 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.59465090594354315121849235360, −7.54318682309951755788940437998, −6.62487848911350690130209658946, −6.58199201198977058359789586536, −6.10059843722885251835990738507, −5.72959658414275361316010104606, −5.45131220258259683041189229071, −5.34584069501345950075555752183, −4.61512916477133518541953882121, −4.46185698311575780337723744245, −4.42578066512789798888497954002, −3.78016847130066560148602255163, −3.07031978392638603524386619879, −2.92638183571929890514770699797, −2.39976234700353712294350703988, −1.71246928549210154885746249595, −1.33977497191023249185908803634, −1.20294161615390550317929615551, 0, 0,
1.20294161615390550317929615551, 1.33977497191023249185908803634, 1.71246928549210154885746249595, 2.39976234700353712294350703988, 2.92638183571929890514770699797, 3.07031978392638603524386619879, 3.78016847130066560148602255163, 4.42578066512789798888497954002, 4.46185698311575780337723744245, 4.61512916477133518541953882121, 5.34584069501345950075555752183, 5.45131220258259683041189229071, 5.72959658414275361316010104606, 6.10059843722885251835990738507, 6.58199201198977058359789586536, 6.62487848911350690130209658946, 7.54318682309951755788940437998, 7.59465090594354315121849235360
