| L(s) = 1 | + (2.17e4 − 3.76e4i)2-s + (2.39e6 + 4.14e6i)3-s + (−6.74e8 − 1.16e9i)4-s + (−9.42e9 + 1.63e10i)5-s + 2.07e11·6-s + (−1.27e12 + 1.26e12i)7-s + (−3.52e13 − 0.00390i)8-s + (−1.14e13 + 1.98e13i)9-s + (4.09e14 + 7.08e14i)10-s + (−8.08e14 − 1.40e15i)11-s + (3.22e15 − 5.58e15i)12-s − 1.38e16·13-s + (1.98e16 + 7.53e16i)14-s + (−9.01e16 + 8i)15-s + (−4.03e17 + 6.99e17i)16-s + (−8.45e16 − 1.46e17i)17-s + ⋯ |
| L(s) = 1 | + (0.937 − 1.62i)2-s + (0.288 + 0.499i)3-s + (−1.25 − 2.17i)4-s + (−0.690 + 1.19i)5-s + 1.08·6-s + (−0.710 + 0.703i)7-s − 2.83·8-s + (−0.166 + 0.288i)9-s + (1.29 + 2.24i)10-s + (−0.642 − 1.11i)11-s + (0.725 − 1.25i)12-s − 0.978·13-s + (0.476 + 1.81i)14-s − 0.797·15-s + (−1.40 + 2.42i)16-s + (−0.121 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.714268492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714268492\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.39e6 - 4.14e6i)T \) |
| 7 | \( 1 + (1.27e12 - 1.26e12i)T \) |
| good | 2 | \( 1 + (-2.17e4 + 3.76e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (9.42e9 - 1.63e10i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (8.08e14 + 1.40e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 + 1.38e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (8.45e16 + 1.46e17i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.66e18 - 4.62e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-3.07e19 + 5.33e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 2.73e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-3.23e21 - 5.61e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-4.02e22 + 6.97e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 - 5.71e22T + 5.89e46T^{2} \) |
| 43 | \( 1 + 4.57e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-8.12e23 + 1.40e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (5.82e24 + 1.00e25i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-4.87e24 - 8.44e24i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-2.38e24 + 4.13e24i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (-1.55e26 - 2.69e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 - 2.89e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (6.70e25 + 1.16e26i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (2.17e27 - 3.77e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 - 5.93e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-1.18e28 + 2.05e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 - 3.54e28T + 4.13e57T^{2} \) |
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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
−11.79145494822072388972010110293, −10.65224628213034646571950148107, −10.08713373338822615736911290548, −8.539139974052640873681168500714, −6.39097018730482755227448981042, −5.06547575871049516151698300118, −3.75566634221862257253971172972, −2.92068379861014080754518462883, −2.43968132466034678896827847420, −0.43340050417828550352279601165,
0.58722064084759710068321196410, 2.84438783522526772140635648017, 4.41381502080863681550336134165, 4.80090718053437567515718346325, 6.48488712028553783253626787575, 7.40996283674888171868268098048, 8.173335458174170531850656636975, 9.470079038183814072786888907460, 12.14607017395228226042433988987, 12.92738347396649283629927761829
