| L(s) = 1 | − 1.73i·3-s + 4.27·5-s − 2.99·9-s − 3.10i·11-s − 2.72·13-s − 7.40i·15-s − 5.09·17-s − 25.7i·19-s + 12.1i·23-s − 6.72·25-s + 5.19i·27-s + 41.0·29-s + 0.172i·31-s − 5.37·33-s − 16.9·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.854·5-s − 0.333·9-s − 0.282i·11-s − 0.209·13-s − 0.493i·15-s − 0.299·17-s − 1.35i·19-s + 0.529i·23-s − 0.269·25-s + 0.192i·27-s + 1.41·29-s + 0.00556i·31-s − 0.162·33-s − 0.457·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.367556523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367556523\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.27T + 25T^{2} \) |
| 11 | \( 1 + 3.10iT - 121T^{2} \) |
| 13 | \( 1 + 2.72T + 169T^{2} \) |
| 17 | \( 1 + 5.09T + 289T^{2} \) |
| 19 | \( 1 + 25.7iT - 361T^{2} \) |
| 23 | \( 1 - 12.1iT - 529T^{2} \) |
| 29 | \( 1 - 41.0T + 841T^{2} \) |
| 31 | \( 1 - 0.172iT - 961T^{2} \) |
| 37 | \( 1 + 16.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 36.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 53.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 24.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 103.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 29.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 76.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 99.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 87.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 68.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 104.T + 9.40e3T^{2} \) |
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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
−8.604844306782414760252602141730, −7.64162974645202659744386370825, −6.83483087784490249596655234651, −6.27090111537046425833960250403, −5.38276061030056758203819579296, −4.66822415348065084963917428030, −3.34305405483823786604794591535, −2.44182202923554376284535595474, −1.56446804686825234270476135346, −0.30851031748906938926625611512,
1.39791534401358633024276037257, 2.40943960575313565103188518209, 3.39602622685667917950425102277, 4.43797930298953191348345294251, 5.11227501335075410962004051213, 6.06682682705351460743788887641, 6.54851624402398205433212861175, 7.73286526046677310164644982897, 8.417401455383566307721229789980, 9.265603532931433304221428707539
