| L(s) = 1 | − 3.74·3-s − 5·5-s + 9.79i·7-s + 5·9-s + 4·11-s − 11.2·13-s + 18.7·15-s + 19.5i·17-s + (−5 − 18.3i)19-s − 36.6i·21-s − 9.79i·23-s + 25·25-s + 14.9·27-s − 36.6i·29-s − 36.6i·31-s + ⋯ |
| L(s) = 1 | − 1.24·3-s − 5-s + 1.39i·7-s + 0.555·9-s + 0.363·11-s − 0.863·13-s + 1.24·15-s + 1.15i·17-s + (−0.263 − 0.964i)19-s − 1.74i·21-s − 0.425i·23-s + 25-s + 0.554·27-s − 1.26i·29-s − 1.18i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.310532 - 0.237173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.310532 - 0.237173i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 19 | \( 1 + (5 + 18.3i)T \) |
| good | 3 | \( 1 + 3.74T + 9T^{2} \) |
| 7 | \( 1 - 9.79iT - 49T^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 + 11.2T + 169T^{2} \) |
| 17 | \( 1 - 19.5iT - 289T^{2} \) |
| 23 | \( 1 + 9.79iT - 529T^{2} \) |
| 29 | \( 1 + 36.6iT - 841T^{2} \) |
| 31 | \( 1 + 36.6iT - 961T^{2} \) |
| 37 | \( 1 + 33.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 36.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 9.79iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 56.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100T + 3.72e3T^{2} \) |
| 67 | \( 1 + 11.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 36.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 19.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 29.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 123.T + 9.40e3T^{2} \) |
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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
−11.20958108545802001847289334698, −10.27738487902452356223994047624, −9.017300450429717843750520681429, −8.219630950644293158063617332677, −6.96958933114405132402230360969, −6.04450655164885400936240255809, −5.17316280045941316953517704142, −4.12808239585791544690296633834, −2.45048175229460902300760867628, −0.26818955682233532850469202854,
0.961598945704208431278308427858, 3.42269381109924929813184693628, 4.51461617531968255687875129590, 5.34082048860877066618963102938, 6.91111301432792060720032354238, 7.16462060014979498520772606038, 8.403563207607293266069982579973, 9.818457813422543242689906054844, 10.66144891725987441394595775501, 11.29716449868319961205310953603
