| L(s) = 1 | − 2-s + 15·4-s + 38·5-s − 18·7-s − 61·8-s − 38·10-s − 242·11-s − 66·13-s + 18·14-s − 721·16-s + 920·17-s − 2.93e3·19-s + 570·20-s + 242·22-s − 5.24e3·23-s + 2.65e3·25-s + 66·26-s − 270·28-s + 1.26e4·29-s + 9.93e3·31-s + 411·32-s − 920·34-s − 684·35-s + 5.99e3·37-s + 2.93e3·38-s − 2.31e3·40-s − 2.42e4·41-s + ⋯ |
| L(s) = 1 | − 0.176·2-s + 0.468·4-s + 0.679·5-s − 0.138·7-s − 0.336·8-s − 0.120·10-s − 0.603·11-s − 0.108·13-s + 0.0245·14-s − 0.704·16-s + 0.772·17-s − 1.86·19-s + 0.318·20-s + 0.106·22-s − 2.06·23-s + 0.850·25-s + 0.0191·26-s − 0.0650·28-s + 2.78·29-s + 1.85·31-s + 0.0709·32-s − 0.136·34-s − 0.0943·35-s + 0.720·37-s + 0.329·38-s − 0.229·40-s − 2.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.933818248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933818248\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 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
−13.27588357768038262609480160578, −12.38160411191282022519429337257, −12.36432890344174913111538734016, −11.70392801160812452455770445476, −10.90671963062090398745723967340, −10.50276957629587416853777440408, −9.899102139833537911626752121192, −9.721381735403661675407767889102, −8.637271772744952013509492775760, −8.283666268948847433762200410167, −7.82696175168471033220401193782, −6.67717527021295441733823544011, −6.37331697449091152897938908404, −6.02620058080176207873274781247, −4.78775395179455972779212828565, −4.50602796201996483100873202284, −3.15855689874682436584742693234, −2.50272710563801659055620116166, −1.79477992483491990780226813003, −0.54147870173894002970784153302,
0.54147870173894002970784153302, 1.79477992483491990780226813003, 2.50272710563801659055620116166, 3.15855689874682436584742693234, 4.50602796201996483100873202284, 4.78775395179455972779212828565, 6.02620058080176207873274781247, 6.37331697449091152897938908404, 6.67717527021295441733823544011, 7.82696175168471033220401193782, 8.283666268948847433762200410167, 8.637271772744952013509492775760, 9.721381735403661675407767889102, 9.899102139833537911626752121192, 10.50276957629587416853777440408, 10.90671963062090398745723967340, 11.70392801160812452455770445476, 12.36432890344174913111538734016, 12.38160411191282022519429337257, 13.27588357768038262609480160578
