L(s) = 1 | + 2i·2-s − 3-s − 2·4-s − 2i·6-s + 2i·7-s − 2·9-s + 2·12-s + (−3 + 2i)13-s − 4·14-s − 4·16-s − 2·17-s − 4i·18-s − 4i·19-s − 2i·21-s − 23-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 0.577·3-s − 4-s − 0.816i·6-s + 0.755i·7-s − 0.666·9-s + 0.577·12-s + (−0.832 + 0.554i)13-s − 1.06·14-s − 16-s − 0.485·17-s − 0.942i·18-s − 0.917i·19-s − 0.436i·21-s − 0.208·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.191891 - 0.633773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.191891 - 0.633773i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (3 - 2i)T \) |
good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−11.93176266694298933058460370640, −11.51797952032722027855627646741, −10.19355003038935935112000501799, −8.867745395695592108036724025108, −8.442749595770913852285471177392, −7.02886080438866008384258629381, −6.47250326800870221202037188489, −5.37167074814700313175886168848, −4.76194707141289287292046691991, −2.61406413337654857103539023048,
0.48137645504867270155404486414, 2.27880256373408315256147343358, 3.56889414762037624882687025841, 4.68951004590417022965209666797, 5.96843779068964231574949881864, 7.20569607189996205962244713118, 8.426577148881573092317124951724, 9.703465973113253747598113452150, 10.36184021706980554193114109300, 11.08018920854494500666756217377