L(s) = 1 | + (1.36 + 0.366i)2-s + (1.67 − 0.448i)3-s + (1.73 + i)4-s + (3.25 + 3.79i)5-s + 2.44·6-s + (−2.26 + 6.62i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (3.05 + 6.37i)10-s + (−1.33 + 2.30i)11-s + (3.34 + 0.896i)12-s + (−9.16 − 9.16i)13-s + (−5.52 + 8.21i)14-s + (7.14 + 4.88i)15-s + (1.99 + 3.46i)16-s + (0.724 + 2.70i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (0.557 − 0.149i)3-s + (0.433 + 0.250i)4-s + (0.651 + 0.758i)5-s + 0.408·6-s + (−0.324 + 0.946i)7-s + (0.249 + 0.250i)8-s + (0.288 − 0.166i)9-s + (0.305 + 0.637i)10-s + (−0.121 + 0.209i)11-s + (0.278 + 0.0747i)12-s + (−0.704 − 0.704i)13-s + (−0.394 + 0.586i)14-s + (0.476 + 0.325i)15-s + (0.124 + 0.216i)16-s + (0.0426 + 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.59836 + 1.11282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.59836 + 1.11282i\) |
\(L(2)\) |
|
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 | \( 1 + (-3.25 - 3.79i)T \) |
| 7 | \( 1 + (2.26 - 6.62i)T \) |
good | 11 | \( 1 + (1.33 - 2.30i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (9.16 + 9.16i)T + 169iT^{2} \) |
| 17 | \( 1 + (-0.724 - 2.70i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-16.5 + 9.57i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-9.36 + 34.9i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 12.4iT - 841T^{2} \) |
| 31 | \( 1 + (-5.33 + 9.24i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-21.9 - 5.88i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 1.17T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-2.70 - 2.70i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (71.7 + 19.2i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-7.83 + 2.09i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (93.6 + 54.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-35.0 - 60.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-4.51 - 16.8i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 66.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (34.9 - 9.37i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-83.6 + 48.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (83.4 + 83.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (125. - 72.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (43.6 - 43.6i)T - 9.40e3iT^{2} \) |
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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
−12.55760893858631848864518334148, −11.47586288752412439285850446561, −10.24156880071596277698558853651, −9.396636286537630179070321048535, −8.129706569916941411619152238517, −6.99789356725184528047330855073, −6.06011508829356483033455722099, −4.93439552849110193684678138822, −3.11385029831396190981097733133, −2.36809055144177624708369469435,
1.49092148665303647052783058366, 3.18231276796121196671395915725, 4.43010381291618910215919552799, 5.46691515301014609774521819154, 6.85741953904947639966345104522, 7.901296846562635591889954169560, 9.404246696440540546040290362213, 9.858969206405755337640736117260, 11.11448956210597084872168610505, 12.25337487320116621091551942213