| L(s) = 1 | + 4·2-s + 12·4-s + 32·8-s − 26·9-s − 30·11-s + 80·16-s − 104·18-s − 120·22-s − 114·23-s − 142·29-s + 192·32-s − 312·36-s + 194·37-s − 198·43-s − 360·44-s − 456·46-s − 556·53-s − 568·58-s + 448·64-s − 170·67-s + 82·71-s − 832·72-s + 776·74-s − 1.57e3·79-s − 53·81-s − 792·86-s − 960·88-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.962·9-s − 0.822·11-s + 5/4·16-s − 1.36·18-s − 1.16·22-s − 1.03·23-s − 0.909·29-s + 1.06·32-s − 1.44·36-s + 0.861·37-s − 0.702·43-s − 1.23·44-s − 1.46·46-s − 1.44·53-s − 1.28·58-s + 7/8·64-s − 0.309·67-s + 0.137·71-s − 1.36·72-s + 1.21·74-s − 2.23·79-s − 0.0727·81-s − 0.993·86-s − 1.16·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 26 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 4142 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8818 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6550 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 57 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 71 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 51490 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 97 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 99510 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 99 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 100014 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 278 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 400650 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 406894 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 85 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 41 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 221366 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 785 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 324774 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 345238 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1791046 T^{2} + p^{6} T^{4} \) |
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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
−8.246651094857675098012120993387, −7.912484591800457621259979940267, −7.53858534340446668549797115129, −7.33724918075770882733142543172, −6.56583834252285405526844080243, −6.41751831479788318700742980565, −5.90248144899013891813519710112, −5.66961259335374195653337802439, −5.25180434163459111149261388921, −4.89364266056276241434141180103, −4.37829935808144785655015068752, −4.06849479800946480144609500025, −3.36358732805281033288940483298, −3.27517305503174713911102835095, −2.47835223493653413725272708868, −2.43839537239734636010331265115, −1.68646483850403124218032289774, −1.15418014944547119267468908904, 0, 0,
1.15418014944547119267468908904, 1.68646483850403124218032289774, 2.43839537239734636010331265115, 2.47835223493653413725272708868, 3.27517305503174713911102835095, 3.36358732805281033288940483298, 4.06849479800946480144609500025, 4.37829935808144785655015068752, 4.89364266056276241434141180103, 5.25180434163459111149261388921, 5.66961259335374195653337802439, 5.90248144899013891813519710112, 6.41751831479788318700742980565, 6.56583834252285405526844080243, 7.33724918075770882733142543172, 7.53858534340446668549797115129, 7.912484591800457621259979940267, 8.246651094857675098012120993387
