| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 3·11-s − 2·14-s + 16-s − 17-s + 3·19-s + 20-s + 3·22-s − 4·23-s + 25-s + 2·28-s + 8·29-s + 31-s − 32-s + 34-s + 2·35-s − 37-s − 3·38-s − 40-s + 8·41-s + 9·43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.688·19-s + 0.223·20-s + 0.639·22-s − 0.834·23-s + 1/5·25-s + 0.377·28-s + 1.48·29-s + 0.179·31-s − 0.176·32-s + 0.171·34-s + 0.338·35-s − 0.164·37-s − 0.486·38-s − 0.158·40-s + 1.24·41-s + 1.37·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.04143488934854, −12.32308248945842, −12.27340830862983, −11.48595511431741, −11.19059010173081, −10.65820620867256, −10.18643263604791, −9.967785913087501, −9.320760800816463, −8.802084037090063, −8.456631244108109, −7.794121251769203, −7.690396383416708, −6.964954943320005, −6.560901341603564, −5.781862246994728, −5.557962338001304, −5.022718064913495, −4.269914002964321, −3.975659558573337, −2.888649537604535, −2.609018770910215, −2.137709552257922, −1.291329889819777, −0.9082565037896350, 0,
0.9082565037896350, 1.291329889819777, 2.137709552257922, 2.609018770910215, 2.888649537604535, 3.975659558573337, 4.269914002964321, 5.022718064913495, 5.557962338001304, 5.781862246994728, 6.560901341603564, 6.964954943320005, 7.690396383416708, 7.794121251769203, 8.456631244108109, 8.802084037090063, 9.320760800816463, 9.967785913087501, 10.18643263604791, 10.65820620867256, 11.19059010173081, 11.48595511431741, 12.27340830862983, 12.32308248945842, 13.04143488934854
