| L(s) = 1 | + 3·5-s + 15·11-s + 51·17-s − 27·19-s − 9·23-s − 19·25-s + 12·29-s + 21·31-s − 31·37-s + 20·43-s + 75·47-s − 57·53-s + 45·55-s − 141·59-s + 141·61-s + 49·67-s + 252·71-s + 45·73-s + 73·79-s + 153·85-s + 99·89-s − 81·95-s + 171·101-s − 123·103-s + 39·107-s − 103·109-s + 156·113-s + ⋯ |
| L(s) = 1 | + 3/5·5-s + 1.36·11-s + 3·17-s − 1.42·19-s − 0.391·23-s − 0.759·25-s + 0.413·29-s + 0.677·31-s − 0.837·37-s + 0.465·43-s + 1.59·47-s − 1.07·53-s + 9/11·55-s − 2.38·59-s + 2.31·61-s + 0.731·67-s + 3.54·71-s + 0.616·73-s + 0.924·79-s + 9/5·85-s + 1.11·89-s − 0.852·95-s + 1.69·101-s − 1.19·103-s + 0.364·107-s − 0.944·109-s + 1.38·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.985814865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.985814865\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 28 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T + 104 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 27 T + 604 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T - 448 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 21 T + 1108 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 31 T - 408 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 75 T + 4084 T^{2} - 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 57 T + 440 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 141 T + 10108 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 141 T + 10348 T^{2} - 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 49 T - 2088 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 45 T + 6004 T^{2} - 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 73 T - 912 T^{2} - 73 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 99 T + 11188 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18050 T^{2} + p^{4} T^{4} \) |
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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
−9.180291625545877732857679871178, −9.180237925833262765716916697646, −8.402141916470308572656059467053, −8.182199092985587484805963053200, −7.75126771900336015797870193365, −7.46578261507165864601002804129, −6.72227594137721823224116770210, −6.53323824291362424042998240078, −6.08477446611837018913437564410, −5.79854655621789369431904677742, −5.19109306446294429836463175854, −5.01769187378074634093187113081, −4.05233214569641472580536053769, −4.01878020400063325518733974964, −3.39396158043668836384956968179, −3.00970886076600830338105095361, −2.00748606097869229359869624353, −1.98395533576885810996430964889, −1.01137810682521955915466730080, −0.71480140545054343947234444467,
0.71480140545054343947234444467, 1.01137810682521955915466730080, 1.98395533576885810996430964889, 2.00748606097869229359869624353, 3.00970886076600830338105095361, 3.39396158043668836384956968179, 4.01878020400063325518733974964, 4.05233214569641472580536053769, 5.01769187378074634093187113081, 5.19109306446294429836463175854, 5.79854655621789369431904677742, 6.08477446611837018913437564410, 6.53323824291362424042998240078, 6.72227594137721823224116770210, 7.46578261507165864601002804129, 7.75126771900336015797870193365, 8.182199092985587484805963053200, 8.402141916470308572656059467053, 9.180237925833262765716916697646, 9.180291625545877732857679871178
