L(s) = 1 | + (−1.36 − 0.366i)2-s + (−1.67 + 0.448i)3-s + (1.73 + i)4-s + (−1.25 + 4.83i)5-s + 2.44·6-s + (5.72 − 4.03i)7-s + (−1.99 − 2i)8-s + (2.59 − 1.50i)9-s + (3.49 − 6.14i)10-s + (−5.44 + 9.42i)11-s + (−3.34 − 0.896i)12-s + (−4.13 − 4.13i)13-s + (−9.29 + 3.41i)14-s + (−0.0638 − 8.66i)15-s + (1.99 + 3.46i)16-s + (−0.489 − 1.82i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.557 + 0.149i)3-s + (0.433 + 0.250i)4-s + (−0.251 + 0.967i)5-s + 0.408·6-s + (0.817 − 0.576i)7-s + (−0.249 − 0.250i)8-s + (0.288 − 0.166i)9-s + (0.349 − 0.614i)10-s + (−0.494 + 0.856i)11-s + (−0.278 − 0.0747i)12-s + (−0.318 − 0.318i)13-s + (−0.663 + 0.243i)14-s + (−0.00425 − 0.577i)15-s + (0.124 + 0.216i)16-s + (−0.0287 − 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.183486 + 0.445682i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.183486 + 0.445682i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 5 | \( 1 + (1.25 - 4.83i)T \) |
| 7 | \( 1 + (-5.72 + 4.03i)T \) |
good | 11 | \( 1 + (5.44 - 9.42i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (4.13 + 4.13i)T + 169iT^{2} \) |
| 17 | \( 1 + (0.489 + 1.82i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (27.1 - 15.6i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (5.85 - 21.8i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 - 35.4iT - 841T^{2} \) |
| 31 | \( 1 + (7.10 - 12.3i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-13.4 - 3.60i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 75.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (5.54 + 5.54i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (89.5 + 23.9i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-73.5 + 19.6i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-39.8 - 22.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-46.7 - 80.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (6.47 + 24.1i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 4.10T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-94.5 + 25.3i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-25.8 + 14.9i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (42.2 + 42.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-71.6 + 41.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (60.0 - 60.0i)T - 9.40e3iT^{2} \) |
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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
−12.14101480940740019178317905073, −11.34118711787347606623355003985, −10.39594230993088967856028251692, −10.08949001651813688900756321297, −8.398000466120613908937145949078, −7.45283854383703956350936311060, −6.67424869308350305345351582778, −5.12294960957611749013609946323, −3.70331199448241574600702504152, −1.92188620600466867750849031218,
0.34628223369835469488790338967, 2.09977682576911866484330384556, 4.49870060241131833515333082480, 5.49644910826214341003396640640, 6.62189250856174654431245403315, 8.175566201568307102309115545797, 8.466050120427528862224252019622, 9.719059630896920821144108599336, 10.97963091500034513435485856453, 11.59281811381526970739662618161