L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 5-s + 4·6-s + 4·8-s + 3·9-s + 2·10-s + 5·11-s + 6·12-s − 2·13-s + 2·15-s + 5·16-s − 17-s + 6·18-s − 5·19-s + 3·20-s + 10·22-s + 3·23-s + 8·24-s + 25-s − 4·26-s + 4·27-s − 29-s + 4·30-s + 6·32-s + 10·33-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s + 1.41·8-s + 9-s + 0.632·10-s + 1.50·11-s + 1.73·12-s − 0.554·13-s + 0.516·15-s + 5/4·16-s − 0.242·17-s + 1.41·18-s − 1.14·19-s + 0.670·20-s + 2.13·22-s + 0.625·23-s + 1.63·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s − 0.185·29-s + 0.730·30-s + 1.06·32-s + 1.74·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(16.58979019\) |
\(L(\frac12)\) |
\(\approx\) |
\(16.58979019\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 132 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 156 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.792089130817399362758465445364, −8.251352455244099970065662024134, −7.73548167499260295173700254230, −7.59584046041812661538650921092, −7.14448322871862587834218805482, −6.75608033364335078939690047310, −6.34529538197202045315364619397, −6.16744775757243342477793843212, −5.74292246227952023492002227871, −5.25722507828402728268668980639, −4.62120120212975969738603863326, −4.42456190823828619421873725895, −4.12674046533058564668665622287, −3.77339413742182160705513686241, −3.20125577427704920430159034803, −2.80037503134609174492594909662, −2.31748354193214101954152805758, −2.12843806786283010744424863656, −1.38638969543504239388096097368, −0.890237578005539139561295283597,
0.890237578005539139561295283597, 1.38638969543504239388096097368, 2.12843806786283010744424863656, 2.31748354193214101954152805758, 2.80037503134609174492594909662, 3.20125577427704920430159034803, 3.77339413742182160705513686241, 4.12674046533058564668665622287, 4.42456190823828619421873725895, 4.62120120212975969738603863326, 5.25722507828402728268668980639, 5.74292246227952023492002227871, 6.16744775757243342477793843212, 6.34529538197202045315364619397, 6.75608033364335078939690047310, 7.14448322871862587834218805482, 7.59584046041812661538650921092, 7.73548167499260295173700254230, 8.251352455244099970065662024134, 8.792089130817399362758465445364