| L(s) = 1 | + 6·3-s − 10·5-s − 14·7-s + 27·9-s + 22·11-s − 24·13-s − 60·15-s + 30·17-s + 32·19-s − 84·21-s − 82·23-s − 38·25-s + 108·27-s − 36·29-s − 112·31-s + 132·33-s + 140·35-s − 48·37-s − 144·39-s − 70·41-s + 40·43-s − 270·45-s − 420·47-s + 147·49-s + 180·51-s − 176·53-s − 220·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 0.603·11-s − 0.512·13-s − 1.03·15-s + 0.428·17-s + 0.386·19-s − 0.872·21-s − 0.743·23-s − 0.303·25-s + 0.769·27-s − 0.230·29-s − 0.648·31-s + 0.696·33-s + 0.676·35-s − 0.213·37-s − 0.591·39-s − 0.266·41-s + 0.141·43-s − 0.894·45-s − 1.30·47-s + 3/7·49-s + 0.494·51-s − 0.456·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 p T + 138 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 p T + 2646 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 24 T + 3990 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 30 T + 9914 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32 T + 11782 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 82 T + 24782 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 36 T - 5698 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 112 T + 53950 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 48 T + 75030 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 70 T + 138930 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 40 T + 104614 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 420 T + 93374 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 176 T + 261110 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 404 T + 451014 T^{2} - 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 156 T + 240846 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 478230 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 814 T + 878046 T^{2} + 814 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 216 T + 499806 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 p T + 1543870 T^{2} + 20 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 464 T + 1195206 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1158 T + 708226 T^{2} + 1158 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1760 T + 2073118 T^{2} + 1760 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.995926815142668447082834777905, −8.639967128334873677608722134978, −8.140672270415479573195030832459, −7.972274782383951708226289312716, −7.33053036747255215976524340031, −7.25353167953266232328208266408, −6.66959409656147213258350491555, −6.33724415184341017828804961206, −5.55741749274650344830593587081, −5.39055774701440019312318478864, −4.51049618450207524276787663175, −4.19882400508894500242291480939, −3.69231612974098368205611929163, −3.48672196550023578231513332047, −2.80131738344867543861765690903, −2.51135983671963256793167652990, −1.59546171071849534203617451820, −1.27477251061760421743995918458, 0, 0,
1.27477251061760421743995918458, 1.59546171071849534203617451820, 2.51135983671963256793167652990, 2.80131738344867543861765690903, 3.48672196550023578231513332047, 3.69231612974098368205611929163, 4.19882400508894500242291480939, 4.51049618450207524276787663175, 5.39055774701440019312318478864, 5.55741749274650344830593587081, 6.33724415184341017828804961206, 6.66959409656147213258350491555, 7.25353167953266232328208266408, 7.33053036747255215976524340031, 7.972274782383951708226289312716, 8.140672270415479573195030832459, 8.639967128334873677608722134978, 8.995926815142668447082834777905
