| L(s) = 1 | + 2.63i·5-s − 12.5i·7-s − 5.79·11-s − 8.78i·13-s + 30.1·17-s − 17.4·19-s + 2.48i·23-s + 18.0·25-s − 26.4i·29-s + 38.0i·31-s + 33.0·35-s − 47.7i·37-s + 53.3·41-s − 30.7·43-s − 16.2i·47-s + ⋯ |
| L(s) = 1 | + 0.527i·5-s − 1.79i·7-s − 0.527·11-s − 0.675i·13-s + 1.77·17-s − 0.916·19-s + 0.108i·23-s + 0.722·25-s − 0.910i·29-s + 1.22i·31-s + 0.945·35-s − 1.29i·37-s + 1.30·41-s − 0.714·43-s − 0.345i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.310403756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310403756\) |
| \(L(2)\) |
|
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 | \( 1 - 2.63iT - 25T^{2} \) |
| 7 | \( 1 + 12.5iT - 49T^{2} \) |
| 11 | \( 1 + 5.79T + 121T^{2} \) |
| 13 | \( 1 + 8.78iT - 169T^{2} \) |
| 17 | \( 1 - 30.1T + 289T^{2} \) |
| 19 | \( 1 + 17.4T + 361T^{2} \) |
| 23 | \( 1 - 2.48iT - 529T^{2} \) |
| 29 | \( 1 + 26.4iT - 841T^{2} \) |
| 31 | \( 1 - 38.0iT - 961T^{2} \) |
| 37 | \( 1 + 47.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 53.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 30.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 16.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 107.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 62.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 19.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.13T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 159.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 39.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 60.5T + 9.40e3T^{2} \) |
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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
−8.271526832426278344554607542035, −7.66436361626107089526038571793, −7.16786050993433394946253378944, −6.29018655611834403874056736869, −5.37205983268757381112161648911, −4.40414199276991553011344071321, −3.57431359839816553664390919910, −2.83574581092471840178514127848, −1.33977251081960084533906478224, −0.33398058121405704267825289028,
1.32835886572665210113283906708, 2.39650553582966454910929954682, 3.18343798200834656412400956516, 4.47833541536337094284746894938, 5.27365487596338419669529598753, 5.83694642027714102655481884274, 6.65034127170768965561316168360, 7.83593772129890191702628385638, 8.388088260004040937566585567846, 9.079944985811262293926791666007
