| L(s) = 1 | + 3·2-s − 3-s + 3·4-s − 3·6-s − 3·8-s − 2·9-s − 3·12-s − 13·16-s + 9·17-s − 6·18-s + 6·23-s + 3·24-s − 6·25-s + 5·27-s − 15·32-s + 27·34-s − 6·36-s + 3·41-s + 18·46-s + 13·48-s − 4·49-s − 18·50-s − 9·51-s − 3·53-s + 15·54-s + 3·64-s + 27·68-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 3/2·4-s − 1.22·6-s − 1.06·8-s − 2/3·9-s − 0.866·12-s − 3.25·16-s + 2.18·17-s − 1.41·18-s + 1.25·23-s + 0.612·24-s − 6/5·25-s + 0.962·27-s − 2.65·32-s + 4.63·34-s − 36-s + 0.468·41-s + 2.65·46-s + 1.87·48-s − 4/7·49-s − 2.54·50-s − 1.26·51-s − 0.412·53-s + 2.04·54-s + 3/8·64-s + 3.27·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1505529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1505529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.215314562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.215314562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 409 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 111 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 159 T^{2} + p^{2} T^{4} \) |
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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
−7.917348781312142135058029833043, −7.28135482330851627854561026930, −6.77012347943565431971774917500, −6.27439671272563323211606656470, −5.86563016999466623599732043789, −5.63113849040683359118259907988, −5.21311234087860608550071794533, −4.95612673290969327045193238797, −4.39463995443385467947697827196, −3.86556318120070831926829869324, −3.48372778655093051893687372300, −2.92629405881545537851228995313, −2.72958093803828564604699444953, −1.57288819464043397737232237803, −0.57073271711852878449856786014,
0.57073271711852878449856786014, 1.57288819464043397737232237803, 2.72958093803828564604699444953, 2.92629405881545537851228995313, 3.48372778655093051893687372300, 3.86556318120070831926829869324, 4.39463995443385467947697827196, 4.95612673290969327045193238797, 5.21311234087860608550071794533, 5.63113849040683359118259907988, 5.86563016999466623599732043789, 6.27439671272563323211606656470, 6.77012347943565431971774917500, 7.28135482330851627854561026930, 7.917348781312142135058029833043
