| L(s) = 1 | − 1.73i·3-s + 1.36·5-s − 1.24i·7-s − 2.99·9-s − 5.79i·11-s + 16.3·13-s − 2.36i·15-s − 5.01·17-s − 26.1i·19-s − 2.15·21-s − 25.1i·23-s − 23.1·25-s + 5.19i·27-s + 32.7·29-s + 1.01i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.272·5-s − 0.177i·7-s − 0.333·9-s − 0.527i·11-s + 1.26·13-s − 0.157i·15-s − 0.294·17-s − 1.37i·19-s − 0.102·21-s − 1.09i·23-s − 0.925·25-s + 0.192i·27-s + 1.13·29-s + 0.0328i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15505 - 1.15505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15505 - 1.15505i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 - 1.36T + 25T^{2} \) |
| 7 | \( 1 + 1.24iT - 49T^{2} \) |
| 11 | \( 1 + 5.79iT - 121T^{2} \) |
| 13 | \( 1 - 16.3T + 169T^{2} \) |
| 17 | \( 1 + 5.01T + 289T^{2} \) |
| 19 | \( 1 + 26.1iT - 361T^{2} \) |
| 23 | \( 1 + 25.1iT - 529T^{2} \) |
| 29 | \( 1 - 32.7T + 841T^{2} \) |
| 31 | \( 1 - 1.01iT - 961T^{2} \) |
| 37 | \( 1 - 14.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 72.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 20.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 111.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 30.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 21.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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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
−11.00857224431501651778148888530, −10.07365103509260759937349088403, −8.781362305727937792581700742534, −8.303076218282563284453057708879, −6.91305688900826819236502404039, −6.29437084102242582304770003234, −5.11018988766870983900749041812, −3.70502391652180962023100693866, −2.33479670649663209301427534178, −0.75991007016035743640294325923,
1.65410291899035953558008272607, 3.30780165044552507882672824335, 4.34420048643453464506313510413, 5.60054704792438069141676967374, 6.38130583196710040844892442648, 7.78424396937742012816586744905, 8.651694888701756370117342390872, 9.661804300754318184217634494625, 10.30436394137592430663631786785, 11.31839175911717333389362691087
