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
−11.59281811381526970739662618161, −10.97963091500034513435485856453, −9.719059630896920821144108599336, −8.466050120427528862224252019622, −8.175566201568307102309115545797, −6.62189250856174654431245403315, −5.49644910826214341003396640640, −4.49870060241131833515333082480, −2.09977682576911866484330384556, −0.34628223369835469488790338967,
1.92188620600466867750849031218, 3.70331199448241574600702504152, 5.12294960957611749013609946323, 6.67424869308350305345351582778, 7.45283854383703956350936311060, 8.398000466120613908937145949078, 10.08949001651813688900756321297, 10.39594230993088967856028251692, 11.34118711787347606623355003985, 12.14101480940740019178317905073