L(s) = 1 | − i·3-s + (0.311 − 2.21i)5-s + i·7-s − 9-s − 2·11-s + 6.42i·13-s + (−2.21 − 0.311i)15-s + 4.42i·17-s − 2.42·19-s + 21-s + 1.37i·23-s + (−4.80 − 1.37i)25-s + i·27-s − 0.755·29-s − 5.18·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.139 − 0.990i)5-s + 0.377i·7-s − 0.333·9-s − 0.603·11-s + 1.78i·13-s + (−0.571 − 0.0803i)15-s + 1.07i·17-s − 0.557·19-s + 0.218·21-s + 0.287i·23-s + (−0.961 − 0.275i)25-s + 0.192i·27-s − 0.140·29-s − 0.931·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8394989210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8394989210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.311 + 2.21i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6.42iT - 13T^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 - 1.37iT - 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 7.61iT - 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 2.75iT - 47T^{2} \) |
| 53 | \( 1 + 9.18iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 2.75iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 1.57iT - 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−9.433792603250466096326390515024, −8.496461260787694253699775505334, −8.283861653990198864381255800090, −7.05752380221728339180934203409, −6.35533725146129395319143987885, −5.47658532352634950038061459233, −4.65112403565697059063266718931, −3.70523218506629136680591324629, −2.18634632881611606317491414288, −1.50625122641719413256357108992,
0.30448326162175073305230041462, 2.35608963455564249349736641983, 3.14715236085892919591777443794, 3.97583967422781439248899551662, 5.24973215276669248711365590235, 5.71039441921052383012876860210, 6.91502716857745439398581482994, 7.51639021233148224979430610947, 8.339966104612306306350756484792, 9.303100583719757210198491439011