| L(s) = 1 | + 42·9-s + 104·13-s − 76·25-s − 824·37-s + 772·49-s − 1.11e3·61-s − 1.68e3·73-s + 1.03e3·81-s − 4.28e3·97-s − 1.43e3·109-s + 4.36e3·117-s − 3.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.02e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 14/9·9-s + 2.21·13-s − 0.607·25-s − 3.66·37-s + 2.25·49-s − 2.33·61-s − 2.70·73-s + 1.41·81-s − 4.48·97-s − 1.25·109-s + 3.45·117-s − 2.70·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.923·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.028356504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028356504\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 14 p T^{2} + p^{6} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 386 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 1798 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 5218 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2186 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 6770 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 48490 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58610 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 206 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 44530 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 150266 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 193822 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 295162 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 98854 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 278 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 191062 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 712366 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 422 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 539090 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1142710 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 1270546 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1070 T + p^{3} T^{2} )^{4} \) |
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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
−8.667010718951912131482805044013, −8.461728742296701997767985863777, −8.286059964108047490627256026812, −7.898803733883064016553794049638, −7.51465416148392328939777908135, −7.23858673239299916003238330386, −7.02358482762530495570447157364, −6.99779084537303488304795442218, −6.28350044840629938410120434809, −6.26134310222814292622176743740, −6.05543776285219732014035139823, −5.46080414388818819171037104086, −5.29537595702398225095889412010, −5.09109086617813649083046122909, −4.48095945985781695147754493775, −4.10167408959360952729375887294, −3.93972410355055321193811451423, −3.81693371236585325364567291328, −3.23471156146629047639679896356, −2.91473677449186139773152097794, −2.37294450124460525331379994650, −1.57332790243082677000825871038, −1.50355090523781583354615996836, −1.26675236284137221341719895586, −0.27973072640983627071548291840,
0.27973072640983627071548291840, 1.26675236284137221341719895586, 1.50355090523781583354615996836, 1.57332790243082677000825871038, 2.37294450124460525331379994650, 2.91473677449186139773152097794, 3.23471156146629047639679896356, 3.81693371236585325364567291328, 3.93972410355055321193811451423, 4.10167408959360952729375887294, 4.48095945985781695147754493775, 5.09109086617813649083046122909, 5.29537595702398225095889412010, 5.46080414388818819171037104086, 6.05543776285219732014035139823, 6.26134310222814292622176743740, 6.28350044840629938410120434809, 6.99779084537303488304795442218, 7.02358482762530495570447157364, 7.23858673239299916003238330386, 7.51465416148392328939777908135, 7.898803733883064016553794049638, 8.286059964108047490627256026812, 8.461728742296701997767985863777, 8.667010718951912131482805044013
