| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s − 3·9-s − 2·11-s + 8·13-s − 4·15-s − 4·17-s + 2·19-s − 4·21-s + 2·23-s + 3·25-s + 14·27-s + 10·29-s − 4·31-s + 4·33-s + 4·35-s + 6·37-s − 16·39-s − 6·41-s − 6·43-s − 6·45-s − 10·47-s − 6·49-s + 8·51-s + 12·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s − 9-s − 0.603·11-s + 2.21·13-s − 1.03·15-s − 0.970·17-s + 0.458·19-s − 0.872·21-s + 0.417·23-s + 3/5·25-s + 2.69·27-s + 1.85·29-s − 0.718·31-s + 0.696·33-s + 0.676·35-s + 0.986·37-s − 2.56·39-s − 0.937·41-s − 0.914·43-s − 0.894·45-s − 1.45·47-s − 6/7·49-s + 1.12·51-s + 1.64·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.654973808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654973808\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22 T + 274 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.169983544291143075918299206095, −7.999259260198570370004698377339, −7.06734150528885614445156808723, −7.01325081512814186162101686853, −6.46813202166870568702930566595, −6.30352836282649072166978479933, −5.88337298809881670747422887315, −5.79031058698162716779319057056, −5.31107040043536446439000990512, −4.94168457505638978245128922134, −4.74607168931526355749533746114, −4.40710556604306588345330764784, −3.53247288496984586766820479884, −3.45131426084314611716155976607, −2.78484428673845277934331133835, −2.63178926483829153760289283684, −1.73936565817257409931342363821, −1.66982505869998123220686515595, −0.799575607015265911563566616355, −0.57202201229147708586567411301,
0.57202201229147708586567411301, 0.799575607015265911563566616355, 1.66982505869998123220686515595, 1.73936565817257409931342363821, 2.63178926483829153760289283684, 2.78484428673845277934331133835, 3.45131426084314611716155976607, 3.53247288496984586766820479884, 4.40710556604306588345330764784, 4.74607168931526355749533746114, 4.94168457505638978245128922134, 5.31107040043536446439000990512, 5.79031058698162716779319057056, 5.88337298809881670747422887315, 6.30352836282649072166978479933, 6.46813202166870568702930566595, 7.01325081512814186162101686853, 7.06734150528885614445156808723, 7.999259260198570370004698377339, 8.169983544291143075918299206095
