L(s) = 1 | + 0.405i·2-s + (−2.95 − 0.545i)3-s + 3.83·4-s + 2.23i·5-s + (0.221 − 1.19i)6-s − 4.55·7-s + 3.18i·8-s + (8.40 + 3.21i)9-s − 0.907·10-s + 6.04i·11-s + (−11.3 − 2.09i)12-s + 3.60·13-s − 1.84i·14-s + (1.21 − 6.59i)15-s + 14.0·16-s + 30.9i·17-s + ⋯ |
L(s) = 1 | + 0.202i·2-s + (−0.983 − 0.181i)3-s + 0.958·4-s + 0.447i·5-s + (0.0368 − 0.199i)6-s − 0.651·7-s + 0.397i·8-s + (0.933 + 0.357i)9-s − 0.0907·10-s + 0.549i·11-s + (−0.942 − 0.174i)12-s + 0.277·13-s − 0.132i·14-s + (0.0812 − 0.439i)15-s + 0.878·16-s + 1.81i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.929774 + 0.773662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.929774 + 0.773662i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.95 + 0.545i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 13 | \( 1 - 3.60T \) |
good | 2 | \( 1 - 0.405iT - 4T^{2} \) |
| 7 | \( 1 + 4.55T + 49T^{2} \) |
| 11 | \( 1 - 6.04iT - 121T^{2} \) |
| 17 | \( 1 - 30.9iT - 289T^{2} \) |
| 19 | \( 1 - 0.389T + 361T^{2} \) |
| 23 | \( 1 - 14.8iT - 529T^{2} \) |
| 29 | \( 1 - 49.2iT - 841T^{2} \) |
| 31 | \( 1 - 30.8T + 961T^{2} \) |
| 37 | \( 1 - 22.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 74.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 103. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 7.55T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 75.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.52T + 5.32e3T^{2} \) |
| 79 | \( 1 + 59.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 41.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 85.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 182.T + 9.40e3T^{2} \) |
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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
−12.41313294634448386231679829095, −11.47608605062154621748015817242, −10.61440697101920268253847476024, −9.962429007783753106628728627489, −8.185197951492547387416768488683, −6.95346483955408686345409614962, −6.44391739706148192480795314583, −5.39219570530274105913660305652, −3.60234466384131275664627652957, −1.77993670808982082749379518655,
0.802191974006879788138725202235, 2.88142839278010796188485495177, 4.51771240543777074614320557262, 5.90231130018192308646916096629, 6.61131996950712195125581199021, 7.79824770015142847754273400068, 9.429101568451484527196936810487, 10.19563593916307689753263822263, 11.35992701113986692356957096906, 11.77978065203740223768957033914