| L(s) = 1 | − 3·3-s + 10·5-s + 14·7-s − 29·9-s + 21·11-s − 23·13-s − 30·15-s + 37·17-s + 102·19-s − 42·21-s − 174·23-s + 75·25-s + 120·27-s − 129·29-s − 320·31-s − 63·33-s + 140·35-s + 44·37-s + 69·39-s − 94·41-s + 198·43-s − 290·45-s − 205·47-s + 147·49-s − 111·51-s − 362·53-s + 210·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s − 1.07·9-s + 0.575·11-s − 0.490·13-s − 0.516·15-s + 0.527·17-s + 1.23·19-s − 0.436·21-s − 1.57·23-s + 3/5·25-s + 0.855·27-s − 0.826·29-s − 1.85·31-s − 0.332·33-s + 0.676·35-s + 0.195·37-s + 0.283·39-s − 0.358·41-s + 0.702·43-s − 0.960·45-s − 0.636·47-s + 3/7·49-s − 0.304·51-s − 0.938·53-s + 0.514·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 38 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 21 T + 2754 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 23 T + 420 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 37 T + 2120 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 102 T + 16246 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 174 T + 31246 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 129 T + 43284 T^{2} + 129 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 320 T + 80510 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 44 T + 72590 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 94 T + 131218 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 198 T + 156478 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 205 T + 218134 T^{2} + 205 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 362 T + 321682 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 512 T + 247366 T^{2} - 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 226 T + 428114 T^{2} - 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 592 T + 547814 T^{2} - 592 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 56 T + 659374 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1316 T + 1013606 T^{2} + 1316 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1299 T + 1390390 T^{2} + 1299 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 420 T - 368394 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 546 T + 343842 T^{2} - 546 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2839 T + 3639120 T^{2} + 2839 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.438117360785666813441240433687, −8.109500915472937561183349338017, −7.75458372601573051599478104357, −7.33796474107514895449921511452, −6.85914733023281121438592031744, −6.57017146323977394068320338365, −5.84542510602971140517700482222, −5.70639225816674917936132250774, −5.34170592382555770066944216010, −5.30835324624605676712741594921, −4.40024143920739431908897292046, −4.18172471078206015588737917575, −3.44360963474572621234341534137, −3.16064465293439383140929665503, −2.40377069501941579448121601395, −2.10126676268422944512480161330, −1.41104998485152332914123091257, −1.14326154184684245422111246115, 0, 0,
1.14326154184684245422111246115, 1.41104998485152332914123091257, 2.10126676268422944512480161330, 2.40377069501941579448121601395, 3.16064465293439383140929665503, 3.44360963474572621234341534137, 4.18172471078206015588737917575, 4.40024143920739431908897292046, 5.30835324624605676712741594921, 5.34170592382555770066944216010, 5.70639225816674917936132250774, 5.84542510602971140517700482222, 6.57017146323977394068320338365, 6.85914733023281121438592031744, 7.33796474107514895449921511452, 7.75458372601573051599478104357, 8.109500915472937561183349338017, 8.438117360785666813441240433687
