| L(s) = 1 | − 5·3-s + 5·5-s + 30·7-s − 27·9-s − 71·11-s + 35·13-s − 25·15-s − 38·19-s − 150·21-s + 5·23-s − 25·25-s + 260·27-s − 155·29-s + 88·31-s + 355·33-s + 150·35-s − 380·37-s − 175·39-s − 142·41-s + 155·43-s − 135·45-s + 455·47-s + 121·49-s + 275·53-s − 355·55-s + 190·57-s − 873·59-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.447·5-s + 1.61·7-s − 9-s − 1.94·11-s + 0.746·13-s − 0.430·15-s − 0.458·19-s − 1.55·21-s + 0.0453·23-s − 1/5·25-s + 1.85·27-s − 0.992·29-s + 0.509·31-s + 1.87·33-s + 0.724·35-s − 1.68·37-s − 0.718·39-s − 0.540·41-s + 0.549·43-s − 0.447·45-s + 1.41·47-s + 0.352·49-s + 0.712·53-s − 0.870·55-s + 0.441·57-s − 1.92·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 5 T + 52 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - p T + 2 p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 30 T + 779 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 71 T + 3716 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 35 T + 3702 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9529 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T - 4378 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 155 T + 19928 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 88 T + 8718 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 380 T + 136878 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 142 T + 135458 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 155 T + 52086 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 455 T + 255038 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 275 T + 191846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 873 T + 591184 T^{2} + 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 445 T + 250812 T^{2} + 445 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 645 T + 674834 T^{2} - 645 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1712 T + 1445258 T^{2} - 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1274 T + 1391022 T^{2} - 1274 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 90 T + 1038382 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 888 T + 1554274 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 710 T + 220818 T^{2} - 710 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.149395680478909961097795249810, −8.509709102810924684702088452231, −8.276986949101237224020711997632, −8.158163011625687449257356833488, −7.50390804945420585177821731749, −7.21877894849149817391165167785, −6.34738715232421568336544774518, −6.26020257736497502214071741744, −5.50771345691728118716779625739, −5.40258529629632205782125147738, −5.08966353095973525499499955594, −4.71329461459773408940567503127, −3.95619542069182722002231512217, −3.43040795104992083915289705758, −2.53370931477470693213653530024, −2.45645241780863011546914379634, −1.63784662278662505139340918555, −1.08335597844511349240491141856, 0, 0,
1.08335597844511349240491141856, 1.63784662278662505139340918555, 2.45645241780863011546914379634, 2.53370931477470693213653530024, 3.43040795104992083915289705758, 3.95619542069182722002231512217, 4.71329461459773408940567503127, 5.08966353095973525499499955594, 5.40258529629632205782125147738, 5.50771345691728118716779625739, 6.26020257736497502214071741744, 6.34738715232421568336544774518, 7.21877894849149817391165167785, 7.50390804945420585177821731749, 8.158163011625687449257356833488, 8.276986949101237224020711997632, 8.509709102810924684702088452231, 9.149395680478909961097795249810
