| L(s) = 1 | + 4·2-s − 2·4-s − 10·5-s − 48·8-s − 40·10-s + 32·11-s − 14·13-s − 92·16-s − 20·17-s − 18·19-s + 20·20-s + 128·22-s + 68·23-s + 75·25-s − 56·26-s + 332·29-s − 66·31-s + 176·32-s − 80·34-s + 18·37-s − 72·38-s + 480·40-s + 152·41-s − 842·43-s − 64·44-s + 272·46-s − 212·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/4·4-s − 0.894·5-s − 2.12·8-s − 1.26·10-s + 0.877·11-s − 0.298·13-s − 1.43·16-s − 0.285·17-s − 0.217·19-s + 0.223·20-s + 1.24·22-s + 0.616·23-s + 3/5·25-s − 0.422·26-s + 2.12·29-s − 0.382·31-s + 0.972·32-s − 0.403·34-s + 0.0799·37-s − 0.307·38-s + 1.89·40-s + 0.578·41-s − 2.98·43-s − 0.219·44-s + 0.871·46-s − 0.657·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.145534242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.145534242\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p^{2} T + 9 p T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 32 T + 30 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 14 T + 3091 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 20 T + 7614 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 18 T + 5607 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 68 T + 24138 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 332 T + 76262 T^{2} - 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 66 T + 58079 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 18 T + 2819 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 152 T + 113850 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 842 T + 333367 T^{2} + 842 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 212 T + 190082 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 368 T + 325338 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 140 T + 346466 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 p T + 580718 T^{2} - 12 p^{4} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1066 T + 760615 T^{2} - 1066 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1208 T + 959606 T^{2} - 1208 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1654 T + 1461763 T^{2} + 1654 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1134 T + 1305519 T^{2} - 1134 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 968 T + 1357022 T^{2} - 968 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 204 T - 767890 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1692 T + 1457670 T^{2} + 1692 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.781052083252697870942203654010, −8.488899747297759386636524048036, −8.171165476791081874683935681412, −7.924197571103961926920479739132, −6.96000057651895779915634137190, −6.94806063581074297503010294357, −6.48557207209414213015463526689, −6.18714173843483552707837169480, −5.36573855743470771324310302708, −5.29253254238450557965051992570, −4.69905005860674101046961254479, −4.61174897831623797293764944408, −3.97994519968574217706562767637, −3.77607776944224225483954918130, −3.28140277247313228239710736056, −2.95141515385465787979954701287, −2.31450123245146563695896025582, −1.56571055140768476810301068236, −0.66425619470808299329277508090, −0.51714090529214921571602484295,
0.51714090529214921571602484295, 0.66425619470808299329277508090, 1.56571055140768476810301068236, 2.31450123245146563695896025582, 2.95141515385465787979954701287, 3.28140277247313228239710736056, 3.77607776944224225483954918130, 3.97994519968574217706562767637, 4.61174897831623797293764944408, 4.69905005860674101046961254479, 5.29253254238450557965051992570, 5.36573855743470771324310302708, 6.18714173843483552707837169480, 6.48557207209414213015463526689, 6.94806063581074297503010294357, 6.96000057651895779915634137190, 7.924197571103961926920479739132, 8.171165476791081874683935681412, 8.488899747297759386636524048036, 8.781052083252697870942203654010
