| L(s) = 1 | + 16·7-s − 64·25-s + 176·31-s − 36·49-s + 64·73-s − 304·79-s + 704·97-s + 112·103-s + 92·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-s + 173-s − 1.02e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 2.55·25-s + 5.67·31-s − 0.734·49-s + 0.876·73-s − 3.84·79-s + 7.25·97-s + 1.08·103-s + 0.760·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.24·169-s + 0.00578·173-s − 5.85·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.74171366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.74171366\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 466 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1664 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2098 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2782 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 4160 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 5810 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4942 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8914 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 176 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.06731952945753248223027695523, −6.06337032267329610217027752168, −5.85428825504225885063987075588, −5.62235373311812430457537750033, −5.39412280022005763078632898776, −4.91752098140652461905448100610, −4.86450393601892199271295197811, −4.60566011930525692929212389140, −4.60220576230541033344308361120, −4.58037593154408979949692461879, −4.08118794736149279403699981437, −4.05133855992274238300234509729, −3.60247648213435426086943048442, −3.23070614190434603255454481273, −3.10570115444951735905462574386, −3.00238115578012267474457659008, −2.57067832050492290453069938379, −2.29976014619415866652152706402, −1.84468951035379798831087471219, −1.81233640217990927207400124198, −1.80027665087945376187880665899, −1.24107438022094981731278304277, −0.809277034049316210801965922397, −0.72569900014832694206620372434, −0.37686662153160130292732900161,
0.37686662153160130292732900161, 0.72569900014832694206620372434, 0.809277034049316210801965922397, 1.24107438022094981731278304277, 1.80027665087945376187880665899, 1.81233640217990927207400124198, 1.84468951035379798831087471219, 2.29976014619415866652152706402, 2.57067832050492290453069938379, 3.00238115578012267474457659008, 3.10570115444951735905462574386, 3.23070614190434603255454481273, 3.60247648213435426086943048442, 4.05133855992274238300234509729, 4.08118794736149279403699981437, 4.58037593154408979949692461879, 4.60220576230541033344308361120, 4.60566011930525692929212389140, 4.86450393601892199271295197811, 4.91752098140652461905448100610, 5.39412280022005763078632898776, 5.62235373311812430457537750033, 5.85428825504225885063987075588, 6.06337032267329610217027752168, 6.06731952945753248223027695523
