| L(s) = 1 | − 2·5-s − 9.79i·7-s + 19.5i·11-s − 21·25-s + 50·29-s − 48.9i·31-s + 19.5i·35-s − 46.9·49-s − 94·53-s − 39.1i·55-s − 117. i·59-s − 50·73-s + 191.·77-s + 146. i·79-s − 97.9i·83-s + ⋯ |
| L(s) = 1 | − 0.400·5-s − 1.39i·7-s + 1.78i·11-s − 0.839·25-s + 1.72·29-s − 1.58i·31-s + 0.559i·35-s − 0.959·49-s − 1.77·53-s − 0.712i·55-s − 1.99i·59-s − 0.684·73-s + 2.49·77-s + 1.86i·79-s − 1.18i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2229946984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2229946984\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2T + 25T^{2} \) |
| 7 | \( 1 + 9.79iT - 49T^{2} \) |
| 11 | \( 1 - 19.5iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 50T + 841T^{2} \) |
| 31 | \( 1 + 48.9iT - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 94T + 2.80e3T^{2} \) |
| 59 | \( 1 + 117. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 146. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 97.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 190T + 9.40e3T^{2} \) |
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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
−8.126977724082118074478874345826, −7.70263637832272259513588172499, −6.97251431510453506260068885846, −6.34395569068220927675544227725, −4.98565696473563310666780974391, −4.34798836266470013911298588525, −3.74754893144402749709364964754, −2.43995065001743789552614979359, −1.31733389704277170242792104640, −0.05661955552560939533794860658,
1.33400402215906214233691604269, 2.74696020645240628597123045673, 3.24359822542426417566104448901, 4.44004974852518002104619470631, 5.45018670817730035360500465058, 5.98103537922693228558033589354, 6.75661426319547008065893387123, 7.933623973047014331030603543449, 8.551321426911397041503909628457, 8.902445189812925091754232267700
