| L(s) = 1 | + (−6.68 − 11.5i)2-s + (−81.7 + 141. i)3-s + (166. − 288. i)4-s + (−961. − 1.66e3i)5-s + 2.18e3·6-s − 1.12e4·8-s + (−3.51e3 − 6.08e3i)9-s + (−1.28e4 + 2.22e4i)10-s + (4.50e4 − 7.81e4i)11-s + (2.72e4 + 4.71e4i)12-s − 3.19e3·13-s + 3.14e5·15-s + (−9.90e3 − 1.71e4i)16-s + (−5.82e4 + 1.00e5i)17-s + (−4.69e4 + 8.12e4i)18-s + (7.12e4 + 1.23e5i)19-s + ⋯ |
| L(s) = 1 | + (−0.295 − 0.511i)2-s + (−0.582 + 1.00i)3-s + (0.325 − 0.564i)4-s + (−0.687 − 1.19i)5-s + 0.687·6-s − 0.975·8-s + (−0.178 − 0.308i)9-s + (−0.406 + 0.703i)10-s + (0.928 − 1.60i)11-s + (0.379 + 0.657i)12-s − 0.0310·13-s + 1.60·15-s + (−0.0378 − 0.0654i)16-s + (−0.169 + 0.292i)17-s + (−0.105 + 0.182i)18-s + (0.125 + 0.217i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0698659 + 0.140936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0698659 + 0.140936i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (6.68 + 11.5i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (81.7 - 141. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (961. + 1.66e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-4.50e4 + 7.81e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 3.19e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + (5.82e4 - 1.00e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-7.12e4 - 1.23e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (6.36e5 + 1.10e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (4.83e6 - 8.37e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-4.33e6 - 7.51e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-5.39e6 - 9.35e6i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.53e7 - 6.12e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (3.20e6 - 5.54e6i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (8.45e7 + 1.46e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.81e7 + 1.00e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (8.00e7 - 1.38e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.44e8 - 4.23e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 8.31e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + (1.04e6 + 1.80e6i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 3.15e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.43557059600693105059491865879, −11.42791581096324827702895102247, −10.70980733387963261796217315389, −9.391420189666179465868301023165, −8.459254652167810434047028422036, −6.15057762377625735058597094764, −4.96282120322762845260584707536, −3.62383549688789993152696294640, −1.23857108781171144512848008199, −0.06685141494532307361870507920,
2.06325993468971701890692992243, 3.78527230608375360527787079346, 6.23373664172257023573478216793, 7.20639568119758748280961175753, 7.53680874011680867720065988536, 9.505771089543085226115563290299, 11.37114486443412467424619013023, 11.90453593031308568302152329245, 12.97275684130275507169836849149, 14.73466478942533618928114126542
