| L(s) = 1 | + 3-s + 4·5-s − 9-s − 2·11-s + 5·13-s + 4·15-s + 2·17-s − 2·19-s + 2·23-s + 2·25-s + 3·29-s + 9·31-s − 2·33-s + 5·39-s + 41-s + 16·43-s − 4·45-s − 11·47-s − 14·49-s + 2·51-s + 4·53-s − 8·55-s − 2·57-s − 4·59-s + 8·61-s + 20·65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 1/3·9-s − 0.603·11-s + 1.38·13-s + 1.03·15-s + 0.485·17-s − 0.458·19-s + 0.417·23-s + 2/5·25-s + 0.557·29-s + 1.61·31-s − 0.348·33-s + 0.800·39-s + 0.156·41-s + 2.43·43-s − 0.596·45-s − 1.60·47-s − 2·49-s + 0.280·51-s + 0.549·53-s − 1.07·55-s − 0.264·57-s − 0.520·59-s + 1.02·61-s + 2.48·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.756605996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.756605996\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 23 T + 270 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−11.60113257813216645399687790474, −11.12636674877091600211447865107, −10.44449225901923892643907893069, −10.33190010772797568951605294828, −9.736755502983405695122385089799, −9.456593206260965572836247007007, −8.794030808201840113568742725296, −8.519144694074493166008228786104, −8.054568304605631569356839700621, −7.51090946812113976891695652584, −6.70943016352112817513708620756, −6.25472220155415915598317827209, −5.79798138527888642819658620913, −5.60346699996532670311983316006, −4.68274507590698707927509197526, −4.18817568019934605305872443906, −3.08721399435752104266901265032, −2.86519380837234695497312762849, −1.99941721652904580146207563496, −1.27333178538438082589803468230,
1.27333178538438082589803468230, 1.99941721652904580146207563496, 2.86519380837234695497312762849, 3.08721399435752104266901265032, 4.18817568019934605305872443906, 4.68274507590698707927509197526, 5.60346699996532670311983316006, 5.79798138527888642819658620913, 6.25472220155415915598317827209, 6.70943016352112817513708620756, 7.51090946812113976891695652584, 8.054568304605631569356839700621, 8.519144694074493166008228786104, 8.794030808201840113568742725296, 9.456593206260965572836247007007, 9.736755502983405695122385089799, 10.33190010772797568951605294828, 10.44449225901923892643907893069, 11.12636674877091600211447865107, 11.60113257813216645399687790474
