| L(s) = 1 | + (0.413 − 0.459i)2-s + (0.169 + 1.60i)4-s + (1.75 + 1.94i)5-s + (2.74 + 1.22i)7-s + (1.80 + 1.31i)8-s + 1.61·10-s + (3.31 + 0.0378i)11-s + (−1.72 − 0.366i)13-s + (1.69 − 0.754i)14-s + (−1.81 + 0.385i)16-s + (−0.5 + 1.53i)17-s + (−4.73 − 3.44i)19-s + (−2.83 + 3.14i)20-s + (1.38 − 1.50i)22-s + (1.73 − 3.00i)23-s + ⋯ |
| L(s) = 1 | + (0.292 − 0.324i)2-s + (0.0845 + 0.804i)4-s + (0.783 + 0.870i)5-s + (1.03 + 0.461i)7-s + (0.639 + 0.464i)8-s + 0.511·10-s + (0.999 + 0.0114i)11-s + (−0.478 − 0.101i)13-s + (0.452 − 0.201i)14-s + (−0.453 + 0.0963i)16-s + (−0.121 + 0.373i)17-s + (−1.08 − 0.789i)19-s + (−0.633 + 0.703i)20-s + (0.296 − 0.321i)22-s + (0.361 − 0.626i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.16352 + 1.11965i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16352 + 1.11965i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-3.31 - 0.0378i)T \) |
| good | 2 | \( 1 + (-0.413 + 0.459i)T + (-0.209 - 1.98i)T^{2} \) |
| 5 | \( 1 + (-1.75 - 1.94i)T + (-0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (-2.74 - 1.22i)T + (4.68 + 5.20i)T^{2} \) |
| 13 | \( 1 + (1.72 + 0.366i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.5 - 1.53i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.73 + 3.44i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.73 + 3.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.08 + 1.81i)T + (19.4 + 21.5i)T^{2} \) |
| 31 | \( 1 + (2.79 + 0.593i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (0.190 - 0.138i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-10.9 + 4.85i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (3.11 + 5.40i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.169 + 1.60i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-2.97 - 9.14i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.07 - 10.2i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (7.68 - 1.63i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.78 + 8.28i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.71 + 5.29i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.61 + 1.90i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-6.33 + 7.03i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.692 + 0.147i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + 0.527T + 89T^{2} \) |
| 97 | \( 1 + (9.39 - 10.4i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−10.67147417001238676853531563890, −9.272218013795379737856550976622, −8.681643262727782590236813970584, −7.64941102970999729201254667860, −6.84682794029720063400958980040, −5.95360066466668744451643455652, −4.76542990582229210870488399764, −3.90269721317006412247022861267, −2.57071487052646845551532726455, −1.97424068157266204135735083077,
1.21328675903441364561798173843, 1.92658839615845194032066616976, 4.03228018569542116856540312999, 4.82851494812782669911539756118, 5.51823800859982978889459485870, 6.40276878907232280614572636694, 7.31870092956907243729565343913, 8.342248062690843574670602609343, 9.373815894490979451416012090388, 9.751952053432800012981679691316
