| L(s) = 1 | − 8·5-s − 40·11-s − 24·13-s − 58·17-s − 26·19-s − 64·23-s + 125·25-s + 124·29-s − 252·31-s − 26·37-s − 12·41-s + 832·43-s − 396·47-s − 450·53-s + 320·55-s + 274·59-s + 576·61-s + 192·65-s + 476·67-s + 896·71-s + 158·73-s + 936·79-s − 1.06e3·83-s + 464·85-s − 390·89-s + 208·95-s + 428·97-s + ⋯ |
| L(s) = 1 | − 0.715·5-s − 1.09·11-s − 0.512·13-s − 0.827·17-s − 0.313·19-s − 0.580·23-s + 25-s + 0.794·29-s − 1.46·31-s − 0.115·37-s − 0.0457·41-s + 2.95·43-s − 1.22·47-s − 1.16·53-s + 0.784·55-s + 0.604·59-s + 1.20·61-s + 0.366·65-s + 0.867·67-s + 1.49·71-s + 0.253·73-s + 1.33·79-s − 1.40·83-s + 0.592·85-s − 0.464·89-s + 0.224·95-s + 0.448·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.733211200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.733211200\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T - 61 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 40 T + 269 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 58 T - 1549 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T - 6183 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 64 T - 8071 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 252 T + 33713 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T - 49977 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 396 T + 52993 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 450 T + 53623 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 274 T - 130303 T^{2} - 274 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 576 T + 104795 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 476 T - 74187 T^{2} - 476 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 448 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 158 T - 364053 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 936 T + 383057 T^{2} - 936 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 530 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 390 T - 552869 T^{2} + 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2} \) |
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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
−9.105236130410637616120455831463, −8.708742635729165716083432075491, −8.149202634123166378105677685223, −8.098082506055592239600883389068, −7.56403956505456797244465115002, −7.25059372457522655395115084363, −6.70060123993165709978008939452, −6.53547911758221244498087788488, −5.85540061850950779536574930553, −5.41894259085668671644672603423, −5.03051243112049013496082168126, −4.66937361502452860721835247861, −3.99438439144310755964852714049, −3.92430849165303129349854604060, −3.10214587634098671619135994730, −2.70212319534457241403283708049, −2.21778901966792836883252487132, −1.72746003118927468358898249699, −0.69162672813409337139823330084, −0.40899398155359271184388571442,
0.40899398155359271184388571442, 0.69162672813409337139823330084, 1.72746003118927468358898249699, 2.21778901966792836883252487132, 2.70212319534457241403283708049, 3.10214587634098671619135994730, 3.92430849165303129349854604060, 3.99438439144310755964852714049, 4.66937361502452860721835247861, 5.03051243112049013496082168126, 5.41894259085668671644672603423, 5.85540061850950779536574930553, 6.53547911758221244498087788488, 6.70060123993165709978008939452, 7.25059372457522655395115084363, 7.56403956505456797244465115002, 8.098082506055592239600883389068, 8.149202634123166378105677685223, 8.708742635729165716083432075491, 9.105236130410637616120455831463
