| L(s) = 1 | + (−0.549 − 1.30i)2-s + (1.01 + 2.89i)3-s + (−1.39 + 1.43i)4-s + (−1.69 + 0.387i)5-s + (3.21 − 2.90i)6-s + (−0.668 + 1.38i)7-s + (2.63 + 1.03i)8-s + (−5.00 + 3.99i)9-s + (1.43 + 2.00i)10-s + (4.29 − 0.483i)11-s + (−5.55 − 2.59i)12-s + (−0.896 − 0.714i)13-s + (2.17 + 0.108i)14-s + (−2.84 − 4.52i)15-s + (−0.0975 − 3.99i)16-s + (5.08 − 5.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.388 − 0.921i)2-s + (0.584 + 1.67i)3-s + (−0.698 + 0.715i)4-s + (−0.759 + 0.173i)5-s + (1.31 − 1.18i)6-s + (−0.252 + 0.524i)7-s + (0.930 + 0.365i)8-s + (−1.66 + 1.33i)9-s + (0.454 + 0.632i)10-s + (1.29 − 0.145i)11-s + (−1.60 − 0.748i)12-s + (−0.248 − 0.198i)13-s + (0.581 + 0.0291i)14-s + (−0.733 − 1.16i)15-s + (−0.0243 − 0.999i)16-s + (1.23 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.771181 + 0.405652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.771181 + 0.405652i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.549 + 1.30i)T \) |
| 29 | \( 1 + (-5.15 + 1.56i)T \) |
| good | 3 | \( 1 + (-1.01 - 2.89i)T + (-2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (1.69 - 0.387i)T + (4.50 - 2.16i)T^{2} \) |
| 7 | \( 1 + (0.668 - 1.38i)T + (-4.36 - 5.47i)T^{2} \) |
| 11 | \( 1 + (-4.29 + 0.483i)T + (10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (0.896 + 0.714i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.08 + 5.08i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3.01 - 1.05i)T + (14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (1.23 + 0.281i)T + (20.7 + 9.97i)T^{2} \) |
| 31 | \( 1 + (4.54 - 7.23i)T + (-13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (-0.587 + 5.21i)T + (-36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (3.03 + 3.03i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1.81 + 1.13i)T + (18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-0.424 - 3.76i)T + (-45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (1.66 + 7.29i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + 1.69iT - 59T^{2} \) |
| 61 | \( 1 + (-3.78 + 1.32i)T + (47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (2.99 + 3.76i)T + (-14.9 + 65.3i)T^{2} \) |
| 71 | \( 1 + (-1.15 + 1.45i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (4.80 - 3.01i)T + (31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (-1.54 + 13.6i)T + (-77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (3.22 + 6.68i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-4.41 - 2.77i)T + (38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (4.80 + 1.68i)T + (75.8 + 60.4i)T^{2} \) |
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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
−14.04042603615884498109647757748, −12.15420757046808258057876657264, −11.56966830820570060414838654004, −10.40373247020855860683421682637, −9.507233397634266233982417368746, −8.902898881537094248326507370204, −7.66905842521836623167631565481, −5.15939583278999860752460965557, −3.84441171759166979863045357705, −3.08278777833720375360045230985,
1.22359673986374952376562855145, 3.86435802997603720683785842883, 6.05076197509114493901146412359, 7.02922822246511990292507258808, 7.78057650154525642318748951684, 8.595697070252513606789747498396, 9.835320296058880335589259167877, 11.68343459809065516652370360743, 12.53970120618052384463258332230, 13.61325342278780070267172484155
