| L(s) = 1 | − 2-s − 18·3-s + 15·4-s − 38·5-s + 18·6-s + 18·7-s − 61·8-s + 243·9-s + 38·10-s − 270·12-s + 66·13-s − 18·14-s + 684·15-s − 721·16-s + 920·17-s − 243·18-s + 2.93e3·19-s − 570·20-s − 324·21-s + 5.24e3·23-s + 1.09e3·24-s + 2.65e3·25-s − 66·26-s − 2.91e3·27-s + 270·28-s + 1.26e4·29-s − 684·30-s + ⋯ |
| L(s) = 1 | − 0.176·2-s − 1.15·3-s + 0.468·4-s − 0.679·5-s + 0.204·6-s + 0.138·7-s − 0.336·8-s + 9-s + 0.120·10-s − 0.541·12-s + 0.108·13-s − 0.0245·14-s + 0.784·15-s − 0.704·16-s + 0.772·17-s − 0.176·18-s + 1.86·19-s − 0.318·20-s − 0.160·21-s + 2.06·23-s + 0.389·24-s + 0.850·25-s − 0.0191·26-s − 0.769·27-s + 0.0650·28-s + 2.78·29-s − 0.138·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.510622038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510622038\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T - 7 p T^{2} + p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 38 T - 1214 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 18 T + 33382 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 66 T - 135542 T^{2} - 66 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 920 T + 3050062 T^{2} - 920 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2932 T + 7100102 T^{2} - 2932 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5246 T + 19699918 T^{2} - 5246 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12600 T + 79348870 T^{2} - 12600 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9936 T + 53013118 T^{2} - 9936 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5996 T + 139663118 T^{2} - 5996 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 24244 T + 337184038 T^{2} + 24244 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 20360 T + 277332086 T^{2} + 20360 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5806 T + 419753950 T^{2} + 5806 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 40770 T + 1224329794 T^{2} - 40770 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18212 T + 455377462 T^{2} - 18212 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11398 T + 1131625826 T^{2} - 11398 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 65368 T + 3722340342 T^{2} - 65368 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 61446 T + 3799408318 T^{2} - 61446 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 53412 T + 4851340822 T^{2} + 53412 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 17122 T + 5872391094 T^{2} + 17122 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14304 T + 6955271542 T^{2} - 14304 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 58140 T + 11297620726 T^{2} + 58140 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 183056 T + 25458744990 T^{2} + 183056 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.27503883911943872189003893894, −10.24263652088769278978183244687, −9.969083751706863906289687919821, −9.850922082120694182400781317185, −8.840268061381146159945479849547, −8.463789527053688707617638045033, −8.101878847874367262115420064693, −7.37063980016741004267402988812, −6.81005746023988529156712146777, −6.71926297369684588405290207032, −6.23887502098069428331586692178, −5.21072357458866822990965327839, −5.00310872892655345383061236010, −4.76034319123985513350227033514, −3.75774503103913722220312301654, −2.98787550659520414410654639560, −2.83381949068248507456223133094, −1.50664132725261592868757402940, −0.893056327116463137385227100393, −0.62559231078408201850357426181,
0.62559231078408201850357426181, 0.893056327116463137385227100393, 1.50664132725261592868757402940, 2.83381949068248507456223133094, 2.98787550659520414410654639560, 3.75774503103913722220312301654, 4.76034319123985513350227033514, 5.00310872892655345383061236010, 5.21072357458866822990965327839, 6.23887502098069428331586692178, 6.71926297369684588405290207032, 6.81005746023988529156712146777, 7.37063980016741004267402988812, 8.101878847874367262115420064693, 8.463789527053688707617638045033, 8.840268061381146159945479849547, 9.850922082120694182400781317185, 9.969083751706863906289687919821, 10.24263652088769278978183244687, 11.27503883911943872189003893894
