| L(s) = 1 | + (−0.667 − 0.178i)2-s + (−2.94 − 0.574i)3-s + (−3.05 − 1.76i)4-s + (1.04 + 4.88i)5-s + (1.86 + 0.909i)6-s + (6.73 − 1.92i)7-s + (3.67 + 3.67i)8-s + (8.34 + 3.38i)9-s + (0.174 − 3.44i)10-s + (13.5 + 7.81i)11-s + (7.97 + 6.93i)12-s + (−6.69 − 6.69i)13-s + (−4.83 + 0.0799i)14-s + (−0.280 − 14.9i)15-s + (5.25 + 9.09i)16-s + (−0.667 + 0.178i)17-s + ⋯ |
| L(s) = 1 | + (−0.333 − 0.0893i)2-s + (−0.981 − 0.191i)3-s + (−0.762 − 0.440i)4-s + (0.209 + 0.977i)5-s + (0.310 + 0.151i)6-s + (0.961 − 0.274i)7-s + (0.459 + 0.459i)8-s + (0.926 + 0.375i)9-s + (0.0174 − 0.344i)10-s + (1.22 + 0.710i)11-s + (0.664 + 0.578i)12-s + (−0.514 − 0.514i)13-s + (−0.345 + 0.00570i)14-s + (−0.0187 − 0.999i)15-s + (0.328 + 0.568i)16-s + (−0.0392 + 0.0105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.801827 + 0.185114i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801827 + 0.185114i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.94 + 0.574i)T \) |
| 5 | \( 1 + (-1.04 - 4.88i)T \) |
| 7 | \( 1 + (-6.73 + 1.92i)T \) |
| good | 2 | \( 1 + (0.667 + 0.178i)T + (3.46 + 2i)T^{2} \) |
| 11 | \( 1 + (-13.5 - 7.81i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (6.69 + 6.69i)T + 169iT^{2} \) |
| 17 | \( 1 + (0.667 - 0.178i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-7.94 - 13.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (5.76 - 21.5i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 - 50.8T + 841T^{2} \) |
| 31 | \( 1 + (-23.6 - 13.6i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-9.69 + 36.1i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 48.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-11.8 + 11.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (13.8 - 51.7i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-1.74 + 0.466i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (12.2 + 7.08i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (10.0 - 5.79i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-98.0 + 26.2i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 28.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (12.5 - 3.37i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (67.3 - 38.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (20.5 - 20.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (59.5 - 34.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (22.9 - 22.9i)T - 9.40e3iT^{2} \) |
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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
−13.85679860398657188677644217237, −12.30499949230427669135026344271, −11.37373940566044712396380809162, −10.35582157482169527025532524211, −9.702174795689770696608180774002, −7.972031615020587518042886442049, −6.82273787278738142948646730552, −5.50429011918907854988144056293, −4.29391063560171438117521339372, −1.46228447922573575041694433307,
0.970454671018086585167185113566, 4.33088881313859992552981846402, 5.02462634998930355357617003799, 6.57635549548022838590132365542, 8.237453373992985044558328137276, 9.044562332187869149972064149241, 10.08512632738013357482287584193, 11.69863088470634773448150421202, 12.07416931102215021069459820240, 13.38105188008307335073916755127
