L(s) = 1 | + (−0.366 − 1.36i)2-s + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (1.65 − 4.71i)5-s + 2.44·6-s + (−6.62 + 2.26i)7-s + (2 + 1.99i)8-s + (−2.59 − 1.50i)9-s + (−7.05 − 0.538i)10-s + (−1.33 − 2.30i)11-s + (−0.896 − 3.34i)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 + (−2.70 − 0.724i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (0.331 − 0.943i)5-s + 0.408·6-s + (−0.946 + 0.324i)7-s + (0.250 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.705 − 0.0538i)10-s + (−0.121 − 0.209i)11-s + (−0.0747 − 0.278i)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 + (−0.159 − 0.0426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00527955 - 0.414744i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00527955 - 0.414744i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 + 1.36i)T \) |
| 3 | \( 1 + (0.448 - 1.67i)T \) |
| 5 | \( 1 + (-1.65 + 4.71i)T \) |
| 7 | \( 1 + (6.62 - 2.26i)T \) |
good | 11 | \( 1 + (1.33 + 2.30i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (9.16 + 9.16i)T + 169iT^{2} \) |
| 17 | \( 1 + (2.70 + 0.724i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (16.5 + 9.57i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (34.9 - 9.36i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 12.4iT - 841T^{2} \) |
| 31 | \( 1 + (-5.33 - 9.24i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (5.88 + 21.9i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 1.17T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-2.70 - 2.70i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-19.2 - 71.7i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (2.09 - 7.83i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 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 | \( 1 + (16.8 + 4.51i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 66.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-9.37 + 34.9i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 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
−11.75077743181462850645468831119, −10.47898787325027655702240158225, −9.745371026243705659767108366205, −9.020026768180159602091609470592, −7.980910486614028258398879738427, −6.18491154579081437121573422503, −5.12790275210769730816035520590, −3.90980521734623947332236629009, −2.42484169538977523137473663643, −0.23404371264569285086575073358,
2.28550421017714030012988194271, 4.01034215075996263453753515869, 5.76865168793090300849699588947, 6.66291324058915958547762069603, 7.22784539732525676984169322946, 8.486101433141024080297400747403, 9.841985975028072141097995343742, 10.35446422328906737043768198005, 11.74052058844367994190161873031, 12.76786267605509706081982387534