| L(s) = 1 | + 1.73i·3-s − 9.08·5-s − 2.99·9-s − 8.80i·11-s − 15.7i·15-s + 2.91·17-s − 17.6i·19-s + 29.5i·23-s + 57.4·25-s − 5.19i·27-s + 48.4·29-s − 38.3i·31-s + 15.2·33-s − 20.4·37-s − 26.9·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.81·5-s − 0.333·9-s − 0.800i·11-s − 1.04i·15-s + 0.171·17-s − 0.926i·19-s + 1.28i·23-s + 2.29·25-s − 0.192i·27-s + 1.67·29-s − 1.23i·31-s + 0.462·33-s − 0.553·37-s − 0.656·41-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3772400627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3772400627\) |
| \(L(2)\) |
|
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 | \( 1 + 9.08T + 25T^{2} \) |
| 11 | \( 1 + 8.80iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 2.91T + 289T^{2} \) |
| 19 | \( 1 + 17.6iT - 361T^{2} \) |
| 23 | \( 1 - 29.5iT - 529T^{2} \) |
| 29 | \( 1 - 48.4T + 841T^{2} \) |
| 31 | \( 1 + 38.3iT - 961T^{2} \) |
| 37 | \( 1 + 20.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 26.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 27.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 62.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 84.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 42.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 52.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 64.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 149. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 0.573iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 10.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 113.T + 9.40e3T^{2} \) |
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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
−8.921856608065678746488713533359, −8.395184463854827562398339984269, −7.66920491049162795458831908432, −7.00650352778592296753245399222, −5.95123021104990982877913028511, −4.94705670332891876028479293632, −4.24387553415347940668954524229, −3.49522375895066060177525853860, −2.81171579167240184761658707562, −0.897730382163920406899777968960,
0.12713160857264115097593821323, 1.29579961334372077832155326038, 2.67872739408659981749653836686, 3.58790847231639284981334313879, 4.41650007776904485662463424224, 5.13040969104617934312059767643, 6.50041927367188181543240216149, 6.99610604459372983326392324348, 7.75644211719127677453688888024, 8.353866221471573343502928061267
