L(s) = 1 | + (−2 − 3.46i)2-s + (4.5 − 7.79i)3-s + (−7.99 + 13.8i)4-s + (12.5 + 21.6i)5-s − 36·6-s + (−40.9 + 122. i)7-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (50 − 86.6i)10-s + (52.9 − 91.6i)11-s + (72 + 124. i)12-s + 416.·13-s + (508. − 103. i)14-s + 225·15-s + (−128 − 221. i)16-s + (−904. + 1.56e3i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (−0.316 + 0.948i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.131 − 0.228i)11-s + (0.144 + 0.249i)12-s + 0.682·13-s + (0.692 − 0.141i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.758 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2396338743\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2396338743\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 + 3.46i)T \) |
| 3 | \( 1 + (-4.5 + 7.79i)T \) |
| 5 | \( 1 + (-12.5 - 21.6i)T \) |
| 7 | \( 1 + (40.9 - 122. i)T \) |
good | 11 | \( 1 + (-52.9 + 91.6i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 416.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (904. - 1.56e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (967. + 1.67e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.42e3 + 4.19e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 2.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-2.28e3 + 3.95e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (348. + 603. i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 8.15e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.32e3 - 4.02e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.59e4 + 2.76e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.48e4 - 2.57e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.50e3 - 2.61e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (5.53e3 - 9.59e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 7.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (3.34e4 - 5.79e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.84e4 - 4.92e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 5.54e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (5.83e4 + 1.01e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 3.45e4T + 8.58e9T^{2} \) |
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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
−10.97924167669362768409270027834, −10.01634237232745905062143890974, −8.737422648539940787721649661191, −8.407222653614307502558691759690, −6.75233887447594351671998781496, −5.96575358871704060355586465922, −4.12679678188131471212762581947, −2.74543852939687236527382275007, −1.85482996052398075882624582190, −0.07475782184884673660145682690,
1.55852815308144100889312999304, 3.57320132135709584020358518128, 4.64872164112814007966150500519, 5.90545023036167703583265188994, 7.07393256102878243325528115893, 8.071510539981663954862104962731, 9.140009371022647451783970890598, 9.885745304625613490490563955057, 10.74805803937258525378013334673, 11.95882732615349237827468531384