| L(s) = 1 | − 4·2-s + 12·4-s − 32·8-s + 80·11-s + 80·16-s − 320·22-s + 136·23-s − 200·25-s + 220·29-s − 192·32-s − 40·37-s − 680·43-s + 960·44-s − 544·46-s + 800·50-s + 1.25e3·53-s − 880·58-s + 448·64-s + 1.08e3·67-s − 840·71-s + 160·74-s − 1.52e3·79-s + 2.72e3·86-s − 2.56e3·88-s + 1.63e3·92-s − 2.40e3·100-s − 5.02e3·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2.19·11-s + 5/4·16-s − 3.10·22-s + 1.23·23-s − 8/5·25-s + 1.40·29-s − 1.06·32-s − 0.177·37-s − 2.41·43-s + 3.28·44-s − 1.74·46-s + 2.26·50-s + 3.25·53-s − 1.99·58-s + 7/8·64-s + 1.96·67-s − 1.40·71-s + 0.251·74-s − 2.16·79-s + 3.41·86-s − 3.10·88-s + 1.84·92-s − 2.39·100-s − 4.60·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.095770324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095770324\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 40 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 344 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9824 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 13590 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 110 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 45470 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 135392 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 340 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 199454 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 628 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 358042 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 388440 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 540 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 420 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 693984 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 760 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 251126 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 81488 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1573296 T^{2} + p^{6} T^{4} \) |
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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
−9.758366027307640953332069230596, −9.671180019111517301039499640022, −8.962615689231983199690085580479, −8.811581198682729746209231294069, −8.410484533161435438734338331949, −8.087069176318155673781024416400, −7.24393562639098688527486782642, −7.11650129153803102762015299885, −6.64968328254312837328815504815, −6.38457337150457888728017283892, −5.67719836724298904910857904410, −5.39020644957797917345643998327, −4.29411479379793709676781048725, −4.27324663551652131896739549068, −3.23146217544154522384284595235, −3.17706791566851427648260946268, −1.96570867801456512209874729673, −1.82623492791353788782857610873, −0.921662543816247780462456153358, −0.61679289556081760323117798203,
0.61679289556081760323117798203, 0.921662543816247780462456153358, 1.82623492791353788782857610873, 1.96570867801456512209874729673, 3.17706791566851427648260946268, 3.23146217544154522384284595235, 4.27324663551652131896739549068, 4.29411479379793709676781048725, 5.39020644957797917345643998327, 5.67719836724298904910857904410, 6.38457337150457888728017283892, 6.64968328254312837328815504815, 7.11650129153803102762015299885, 7.24393562639098688527486782642, 8.087069176318155673781024416400, 8.410484533161435438734338331949, 8.811581198682729746209231294069, 8.962615689231983199690085580479, 9.671180019111517301039499640022, 9.758366027307640953332069230596
