| L(s) = 1 | − 3-s − 5-s − 1.63i·7-s + 9-s − 1.02i·11-s − 1.63i·13-s + 15-s + 3.31·17-s + (4.31 + 0.616i)19-s + 1.63i·21-s + 8.09i·23-s + 25-s − 27-s − 1.02i·29-s − 6.30·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.619i·7-s + 0.333·9-s − 0.308i·11-s − 0.454i·13-s + 0.258·15-s + 0.804·17-s + (0.989 + 0.141i)19-s + 0.357i·21-s + 1.68i·23-s + 0.200·25-s − 0.192·27-s − 0.189i·29-s − 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.345395145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345395145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-4.31 - 0.616i)T \) |
| good | 7 | \( 1 + 1.63iT - 7T^{2} \) |
| 11 | \( 1 + 1.02iT - 11T^{2} \) |
| 13 | \( 1 + 1.63iT - 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 23 | \( 1 - 8.09iT - 23T^{2} \) |
| 29 | \( 1 + 1.02iT - 29T^{2} \) |
| 31 | \( 1 + 6.30T + 31T^{2} \) |
| 37 | \( 1 - 9.11iT - 37T^{2} \) |
| 41 | \( 1 + 2.25iT - 41T^{2} \) |
| 43 | \( 1 - 9.11iT - 43T^{2} \) |
| 47 | \( 1 + 6.85iT - 47T^{2} \) |
| 53 | \( 1 + 8.09iT - 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 - 0.989T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 - 5.22iT - 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.039541230790569736320935222199, −7.58928300679870304181948758974, −6.99932558034886080422295636961, −6.00999253524596650777242094759, −5.39519313063153846780594766293, −4.69774179592031188033722466169, −3.58314662754336707994178878673, −3.25149379641849326733817922861, −1.62269645581423776388055752132, −0.70632541317693189676312896758,
0.66013527500673227512376064884, 1.91941063877319390263066336520, 2.92230921525461363192249061481, 3.90530412683996598864837199260, 4.67175610996637330056584487220, 5.46608465487536351061739965861, 6.03672566031754810033255819734, 7.00752767994055434456627787293, 7.46661862982695984432640617279, 8.327111663331528540895861201684
