| L(s) = 1 | + 10.2·5-s + 41.1·7-s + 169.·11-s + 103. i·13-s + 100.·17-s + (348. − 94.0i)19-s + 299.·23-s − 520.·25-s − 602. i·29-s + 1.26e3i·31-s + 420.·35-s + 1.25e3i·37-s − 1.30e3i·41-s + 628.·43-s + 142.·47-s + ⋯ |
| L(s) = 1 | + 0.408·5-s + 0.839·7-s + 1.40·11-s + 0.610i·13-s + 0.347·17-s + (0.965 − 0.260i)19-s + 0.565·23-s − 0.833·25-s − 0.715i·29-s + 1.32i·31-s + 0.342·35-s + 0.913i·37-s − 0.776i·41-s + 0.339·43-s + 0.0643·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.062379315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.062379315\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-348. + 94.0i)T \) |
| good | 5 | \( 1 - 10.2T + 625T^{2} \) |
| 7 | \( 1 - 41.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 169.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 103. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 100.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 299.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 602. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.26e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.25e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.30e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 628.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 142.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 2.88e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.07e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.89e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.60e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.34e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 946.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.41e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 7.76e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.39e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.53e4iT - 8.85e7T^{2} \) |
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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
−9.769039545775748392869383966434, −9.159689003016154797377491853432, −8.272224437039143138445406820417, −7.23597307895032751998248473078, −6.42778495511075481908052516019, −5.37649113227317311339287578956, −4.45326732539632828592727411213, −3.37897201138717862982128699116, −1.91079534122912203342748933481, −1.05986524558820741563069326306,
0.914894312095336643360286970451, 1.84631997273912617316786119150, 3.26522642128670357072303632669, 4.32894812410891233901978493095, 5.41105964482755495874742739398, 6.18003021471276234091841077186, 7.33841435257960578158717509998, 8.055694366921851258557942336163, 9.154227656457213478151326936157, 9.697293796684225902722204331654
