L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s − i·8-s + 9-s + 10-s − i·11-s + 12-s + i·15-s + 16-s + 17-s + i·18-s − i·19-s + i·20-s + ⋯ |
L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s − i·8-s + 9-s + 10-s − i·11-s + 12-s + i·15-s + 16-s + 17-s + i·18-s − i·19-s + i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5971313470 - 0.08841648564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5971313470 - 0.08841648564i\) |
\(L(1)\) |
\(\approx\) |
\(0.6936697529 + 0.08586778093i\) |
\(L(1)\) |
\(\approx\) |
\(0.6936697529 + 0.08586778093i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 \) |
| 47 | \( 1 \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.19793506851812591959544143602, −29.500670805264426120866868419715, −28.458076096948351153722869128646, −27.56320634809121601362079656630, −26.695057678809649452004603758413, −25.37419681944931730115046903627, −23.44991048913962279752206623899, −22.9707297400933119619317504048, −21.99852271909838633126610321644, −21.118984524047257926440389980934, −19.716559356106776123535234695356, −18.45489692930169248311723676399, −17.97616187487882684676454836399, −16.71709243325565039175562974677, −15.08237264085776506170646268615, −13.89135232336882052480834930487, −12.37223770369539347107106395980, −11.74126716824073202979038599380, −10.34471011932796142454623721514, −9.96142290485052827447587995496, −7.81464921538810475779482282019, −6.32476991024858219773884900865, −4.897970026521705576749466222211, −3.487920297345105007216077359806, −1.74688801146784934695314227534,
0.79098535481316357194219286347, 4.091568173825023058291045911171, 5.29707701831797870143384365811, 6.1096988193255291963287116219, 7.607890289957269577787573349911, 8.84257983275624655111093410830, 10.115640289233802259487868813382, 11.75881946912525345280345906824, 12.86772167826348890266508981370, 13.91437512458049024543337876256, 15.61301557309509256390888583366, 16.373962799470795582112781293824, 17.13092713157679422175474281953, 18.158411438232013043640726409646, 19.37982069651019617893735395766, 21.19967313241630427958440482778, 22.05438338091677543734697544025, 23.32367840869749348859247473429, 24.04842845993512499830329085666, 24.76610368572535560130704637034, 26.14292774101134690782756064314, 27.40200212782066892597448644438, 28.01677794982949136426300341850, 29.08231883441934029483788916977, 30.36235256379769588690836441371