| L(s) = 1 | + 5·3-s − 17·5-s + 14·7-s − 21·9-s + 22·11-s − 86·13-s − 85·15-s − 66·17-s + 136·19-s + 70·21-s − 17·23-s + 95·25-s − 200·27-s + 124·29-s + 71·31-s + 110·33-s − 238·35-s − 135·37-s − 430·39-s − 618·41-s − 604·43-s + 357·45-s − 2·47-s + 147·49-s − 330·51-s + 152·53-s − 374·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 1.52·5-s + 0.755·7-s − 7/9·9-s + 0.603·11-s − 1.83·13-s − 1.46·15-s − 0.941·17-s + 1.64·19-s + 0.727·21-s − 0.154·23-s + 0.759·25-s − 1.42·27-s + 0.794·29-s + 0.411·31-s + 0.580·33-s − 1.14·35-s − 0.599·37-s − 1.76·39-s − 2.35·41-s − 2.14·43-s + 1.18·45-s − 0.00620·47-s + 3/7·49-s − 0.906·51-s + 0.393·53-s − 0.916·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 5 T + 46 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 17 T + 194 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 86 T + 6186 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 66 T + 1282 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 136 T + 18114 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 17 T + 18122 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 124 T + 19790 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 71 T + 59688 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 135 T + 101744 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 618 T + 203170 T^{2} + 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 604 T + 211686 T^{2} + 604 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 48226 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 152 T - 118042 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 343 T + 251714 T^{2} - 343 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 380 T + 7614 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1027 T + 851514 T^{2} + 1027 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1169 T + 1028606 T^{2} - 1169 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 94 T + 755106 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 842 T + 1127694 T^{2} + 842 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1486 T + 1670486 T^{2} - 1486 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 361 T + 1240724 T^{2} + 361 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 225 T + 1794896 T^{2} - 225 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.965789640618317442242401539433, −8.596240282729508514910398704147, −8.194614939414958622820808412601, −8.183394936218209261580222839283, −7.47090262732493261806893103095, −7.36086694228590903178907474779, −6.64834927222021733116547255860, −6.61466513724391250280673592379, −5.47569522684061874972217817267, −5.26577134860883502587812435887, −4.74939898267623058648779762062, −4.46256433401156783300579823659, −3.57989679136885331387883811667, −3.54453773665103391522966030240, −2.91553962195133858767669820248, −2.40806162750964078785910804729, −1.86128630814384674719530421905, −1.06027715211277279575475315198, 0, 0,
1.06027715211277279575475315198, 1.86128630814384674719530421905, 2.40806162750964078785910804729, 2.91553962195133858767669820248, 3.54453773665103391522966030240, 3.57989679136885331387883811667, 4.46256433401156783300579823659, 4.74939898267623058648779762062, 5.26577134860883502587812435887, 5.47569522684061874972217817267, 6.61466513724391250280673592379, 6.64834927222021733116547255860, 7.36086694228590903178907474779, 7.47090262732493261806893103095, 8.183394936218209261580222839283, 8.194614939414958622820808412601, 8.596240282729508514910398704147, 8.965789640618317442242401539433
