| L(s) = 1 | + 3-s + 5-s − 4·7-s − 3·11-s + 2·13-s + 15-s + 7·19-s − 4·21-s + 5·23-s − 27-s − 6·31-s − 3·33-s − 4·35-s − 3·37-s + 2·39-s − 6·41-s + 16·43-s + 47-s + 9·49-s − 5·53-s − 3·55-s + 7·57-s + 4·59-s + 8·61-s + 2·65-s + 5·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s − 0.904·11-s + 0.554·13-s + 0.258·15-s + 1.60·19-s − 0.872·21-s + 1.04·23-s − 0.192·27-s − 1.07·31-s − 0.522·33-s − 0.676·35-s − 0.493·37-s + 0.320·39-s − 0.937·41-s + 2.43·43-s + 0.145·47-s + 9/7·49-s − 0.686·53-s − 0.404·55-s + 0.927·57-s + 0.520·59-s + 1.02·61-s + 0.248·65-s + 0.601·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.953371714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.953371714\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−10.18181195553923986417009143545, −9.970557906647177490074991949555, −9.542330945551126381084793250484, −9.229187180127612862549255510051, −8.729771309056488970559466485851, −8.546673101914804046324336293352, −7.65884994490728300491640142433, −7.51385722903364949545006383144, −7.02830227740962485191072781350, −6.57805653316657068181928750754, −6.00688050514278777277832728788, −5.55560437689190182492672982133, −5.30107715318461024438352298522, −4.61720156783402404379653251923, −3.70711171926691533058155850949, −3.55211196307285369492281623819, −2.82134422501107753721185532077, −2.63034589372546761181501476025, −1.65672274062896908296327704054, −0.66766699931631107212832506252,
0.66766699931631107212832506252, 1.65672274062896908296327704054, 2.63034589372546761181501476025, 2.82134422501107753721185532077, 3.55211196307285369492281623819, 3.70711171926691533058155850949, 4.61720156783402404379653251923, 5.30107715318461024438352298522, 5.55560437689190182492672982133, 6.00688050514278777277832728788, 6.57805653316657068181928750754, 7.02830227740962485191072781350, 7.51385722903364949545006383144, 7.65884994490728300491640142433, 8.546673101914804046324336293352, 8.729771309056488970559466485851, 9.229187180127612862549255510051, 9.542330945551126381084793250484, 9.970557906647177490074991949555, 10.18181195553923986417009143545
