| L(s) = 1 | − 2.82i·5-s + (−3 − 1.41i)11-s + 4.24i·13-s − 6·17-s + 4.24i·19-s − 1.41i·23-s − 3.00·25-s + 2·31-s + 10·37-s − 6·41-s − 12.7i·43-s − 9.89i·47-s + 7·49-s + 5.65i·53-s + (−4.00 + 8.48i)55-s + ⋯ |
| L(s) = 1 | − 1.26i·5-s + (−0.904 − 0.426i)11-s + 1.17i·13-s − 1.45·17-s + 0.973i·19-s − 0.294i·23-s − 0.600·25-s + 0.359·31-s + 1.64·37-s − 0.937·41-s − 1.94i·43-s − 1.44i·47-s + 49-s + 0.777i·53-s + (−0.539 + 1.14i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242698422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242698422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 15.5iT - 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.398851285630526314090886224641, −7.38356510513658713427488831196, −6.72934577407889125906309352202, −5.79069102123679957078895673105, −5.28869751120526432200747245580, −4.31313465075571511389104274236, −4.07993841732859832515307435884, −2.66010179647600734173536061927, −1.89950673128867001433390172872, −0.806551829878677410317873596074,
0.39409512457413729087875034424, 2.01632810851628553342352113740, 2.83646576687708102922253280684, 3.19844324177025084449694764817, 4.51729202347086437545146482302, 4.96238023448040029533516229693, 6.16235194739329730265868363000, 6.42213749975465502838848729785, 7.44622708153787062098797499187, 7.70431465069360961288301291759
