| L(s) = 1 | − 7-s − 6·11-s − 2·13-s − 8·19-s − 6·23-s + 5·25-s + 6·29-s − 8·31-s + 4·37-s + 12·41-s + 4·43-s + 12·47-s + 12·53-s + 10·61-s − 8·67-s − 12·71-s − 20·73-s + 6·77-s + 4·79-s − 12·83-s − 24·89-s + 2·91-s + 10·97-s − 12·101-s − 8·103-s + 12·107-s + 28·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s − 0.554·13-s − 1.83·19-s − 1.25·23-s + 25-s + 1.11·29-s − 1.43·31-s + 0.657·37-s + 1.87·41-s + 0.609·43-s + 1.75·47-s + 1.64·53-s + 1.28·61-s − 0.977·67-s − 1.42·71-s − 2.34·73-s + 0.683·77-s + 0.450·79-s − 1.31·83-s − 2.54·89-s + 0.209·91-s + 1.01·97-s − 1.19·101-s − 0.788·103-s + 1.16·107-s + 2.68·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6346130780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6346130780\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.954083983417968950778595705529, −8.714856379864109155626557695190, −8.691708748413608990135737158582, −8.025460003388224801738524162685, −7.58700247885894869043234030820, −7.34087334667278035791086019937, −7.06485535008402857043182799627, −6.45319460686884828453882081888, −5.94208133767132300212196196801, −5.69729846878226190519614019299, −5.47005332806805922470972451189, −4.66826700627741717419464487389, −4.35508326462730648850119497692, −4.15706293341624450597361385327, −3.43994453101223211628149269579, −2.61454819182830619596687862796, −2.59775249824722536351258407399, −2.19555793939704123652728816113, −1.20751825795538845446611096919, −0.28194619615959937496546034655,
0.28194619615959937496546034655, 1.20751825795538845446611096919, 2.19555793939704123652728816113, 2.59775249824722536351258407399, 2.61454819182830619596687862796, 3.43994453101223211628149269579, 4.15706293341624450597361385327, 4.35508326462730648850119497692, 4.66826700627741717419464487389, 5.47005332806805922470972451189, 5.69729846878226190519614019299, 5.94208133767132300212196196801, 6.45319460686884828453882081888, 7.06485535008402857043182799627, 7.34087334667278035791086019937, 7.58700247885894869043234030820, 8.025460003388224801738524162685, 8.691708748413608990135737158582, 8.714856379864109155626557695190, 8.954083983417968950778595705529
