| L(s) = 1 | + 6.26i·5-s − 2.64i·7-s + 9.80·11-s − 2.41i·13-s − 6.89·17-s − 2.77·19-s − 42.8i·23-s − 14.2·25-s − 37.3i·29-s − 7.16i·31-s + 16.5·35-s + 0.202i·37-s − 63.5·41-s + 35.3·43-s + 37.9i·47-s + ⋯ |
| L(s) = 1 | + 1.25i·5-s − 0.377i·7-s + 0.891·11-s − 0.185i·13-s − 0.405·17-s − 0.146·19-s − 1.86i·23-s − 0.571·25-s − 1.28i·29-s − 0.231i·31-s + 0.473·35-s + 0.00547i·37-s − 1.54·41-s + 0.821·43-s + 0.806i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.738519721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.738519721\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 6.26iT - 25T^{2} \) |
| 11 | \( 1 - 9.80T + 121T^{2} \) |
| 13 | \( 1 + 2.41iT - 169T^{2} \) |
| 17 | \( 1 + 6.89T + 289T^{2} \) |
| 19 | \( 1 + 2.77T + 361T^{2} \) |
| 23 | \( 1 + 42.8iT - 529T^{2} \) |
| 29 | \( 1 + 37.3iT - 841T^{2} \) |
| 31 | \( 1 + 7.16iT - 961T^{2} \) |
| 37 | \( 1 - 0.202iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 63.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 35.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 104.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 31.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 23.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 69.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 19.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 5.11T + 6.88e3T^{2} \) |
| 89 | \( 1 - 17.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 12.4T + 9.40e3T^{2} \) |
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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
−8.812736422240899109221249756657, −8.090481962339880570898925093431, −7.09210429634157057163054606661, −6.61247423612278299935705928451, −5.97549946111809402552259088774, −4.64049487744299484754801900747, −3.89084984441262590094057850663, −2.93489128069278887996081964948, −2.03801334447952913018176326746, −0.49132977639356202104552956938,
1.06994358405321471640389013242, 1.86752616822704202667864214092, 3.33143768331963988976705474720, 4.21018656339660292420889202337, 5.09857317613918262386551160561, 5.69723240916291170619235197610, 6.73936255979291677702121983831, 7.50719497156922482672194610609, 8.652682717082921943122932160496, 8.876178492335610550623655696261
