| L(s) = 1 | + 32·5-s − 296·13-s − 576·17-s − 420·25-s − 3.48e3·29-s + 1.32e3·37-s − 6.84e3·41-s + 3.04e3·49-s − 1.10e4·53-s + 424·61-s − 9.47e3·65-s + 3.00e3·73-s − 1.84e4·85-s + 384·89-s + 1.11e4·97-s + 2.91e3·101-s + 1.86e4·109-s − 2.17e3·113-s + 2.73e4·121-s − 2.06e4·125-s + ⋯ |
| L(s) = 1 | + 1.27·5-s − 1.75·13-s − 1.99·17-s − 0.671·25-s − 4.14·29-s + 0.964·37-s − 4.07·41-s + 1.26·49-s − 3.93·53-s + 0.113·61-s − 2.24·65-s + 0.562·73-s − 2.55·85-s + 0.0484·89-s + 1.18·97-s + 0.285·101-s + 1.56·109-s − 0.170·113-s + 1.86·121-s − 1.32·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3607725911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3607725911\) |
| \(L(3)\) |
|
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 | $D_{4}$ | \( ( 1 - 16 T + 594 T^{2} - 16 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3044 T^{2} + 9238086 T^{4} - 3044 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 27332 T^{2} + 426733638 T^{4} - 27332 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 148 T + 16518 T^{2} + 148 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 288 T + 184898 T^{2} + 288 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 497092 T^{2} + 265140438 p^{2} T^{4} - 497092 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 257156 T^{2} + 145975182726 T^{4} - 257156 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1744 T + 2053266 T^{2} + 1744 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2943332 T^{2} + 3834092018118 T^{4} - 2943332 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 660 T + 2705222 T^{2} - 660 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 3424 T + 8095746 T^{2} + 3424 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6217796 T^{2} + 20023638876486 T^{4} - 6217796 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3265276 T^{2} + 3102160162566 T^{4} + 3265276 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 5520 T + 19769042 T^{2} + 5520 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4500932 T^{2} - 162959688247482 T^{4} - 4500932 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 212 T + 27288198 T^{2} - 212 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 15078212 T^{2} + 51214632004038 T^{4} - 15078212 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 24936836 T^{2} + 605069020552326 T^{4} - 24936836 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 1500 T + 57174662 T^{2} - 1500 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 45515876 T^{2} + 513196620838086 T^{4} - 45515876 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 134137412 T^{2} + 8408135231992518 T^{4} - 134137412 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 192 T + 30096578 T^{2} - 192 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 5564 T + 18910086 T^{2} - 5564 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−7.76467404428422403238288071189, −7.66062358485390258266372420408, −7.58096939790848059834986185650, −6.86926657478589205767451599783, −6.85739779776301249186845288579, −6.75640451426444804609067597804, −6.45410571948803631077412711689, −5.82552501759511521932664285985, −5.80134369573157919111277230145, −5.49297799211606867820126738694, −5.48056167083310275764083947850, −4.74704185365126160049361069711, −4.67030459191370505932793425057, −4.54469870793056852772992193589, −4.13663694203822726683274146750, −3.44617764045850277396434860042, −3.30162559308894551896030383770, −3.30125473775552658026969136758, −2.37147043850594435187442654792, −2.12527486376191145180852029262, −1.96320596863848536947230721365, −1.87709325972358840273433614262, −1.36906112092296378067227337118, −0.42525784779587062547055438587, −0.12628598304834161955609829306,
0.12628598304834161955609829306, 0.42525784779587062547055438587, 1.36906112092296378067227337118, 1.87709325972358840273433614262, 1.96320596863848536947230721365, 2.12527486376191145180852029262, 2.37147043850594435187442654792, 3.30125473775552658026969136758, 3.30162559308894551896030383770, 3.44617764045850277396434860042, 4.13663694203822726683274146750, 4.54469870793056852772992193589, 4.67030459191370505932793425057, 4.74704185365126160049361069711, 5.48056167083310275764083947850, 5.49297799211606867820126738694, 5.80134369573157919111277230145, 5.82552501759511521932664285985, 6.45410571948803631077412711689, 6.75640451426444804609067597804, 6.85739779776301249186845288579, 6.86926657478589205767451599783, 7.58096939790848059834986185650, 7.66062358485390258266372420408, 7.76467404428422403238288071189
