| L(s) = 1 | + (−205. + 355. i)3-s + (−41.2 − 71.4i)5-s + (4.33e4 + 1.00e4i)7-s + (4.34e3 + 7.53e3i)9-s + (−1.60e5 + 2.78e5i)11-s + 8.11e5·13-s + 3.38e4·15-s + (2.13e6 − 3.69e6i)17-s + (7.49e6 + 1.29e7i)19-s + (−1.24e7 + 1.33e7i)21-s + (−3.29e6 − 5.70e6i)23-s + (2.44e7 − 4.22e7i)25-s − 7.62e7·27-s + 1.83e8·29-s + (7.42e7 − 1.28e8i)31-s + ⋯ |
| L(s) = 1 | + (−0.487 + 0.844i)3-s + (−0.00590 − 0.0102i)5-s + (0.974 + 0.225i)7-s + (0.0245 + 0.0425i)9-s + (−0.300 + 0.521i)11-s + 0.606·13-s + 0.0115·15-s + (0.364 − 0.630i)17-s + (0.694 + 1.20i)19-s + (−0.665 + 0.712i)21-s + (−0.106 − 0.184i)23-s + (0.499 − 0.865i)25-s − 1.02·27-s + 1.66·29-s + (0.465 − 0.806i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.357212507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357212507\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-4.33e4 - 1.00e4i)T \) |
| good | 3 | \( 1 + (205. - 355. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (41.2 + 71.4i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.60e5 - 2.78e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 - 8.11e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-2.13e6 + 3.69e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-7.49e6 - 1.29e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (3.29e6 + 5.70e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.83e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-7.42e7 + 1.28e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (2.93e8 + 5.07e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 7.84e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.21e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (7.11e8 + 1.23e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.68e8 - 2.92e8i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-5.07e9 + 8.79e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.96e9 - 3.39e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (8.62e9 - 1.49e10i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.18e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (7.04e9 - 1.21e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-3.25e9 - 5.63e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 1.57e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-4.14e10 - 7.18e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 6.77e10T + 7.15e21T^{2} \) |
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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
−11.55112403212801739790677746363, −10.57433944590421199915564281798, −9.840461477526446157403119200713, −8.476675260238620303053752546109, −7.48193673821616158682156286726, −5.84895018069870514769358553248, −4.93001472125719110505554650671, −4.02734289453823078555902281649, −2.35463181134387830527097830658, −0.937392559326585423036217912499,
0.798017277233891283971330581998, 1.41600118496215426256158412712, 3.06225253643972367116478126532, 4.63567466582923510517594542975, 5.81289151144505813089437098264, 6.89778189115584620242254538702, 7.88529140455891821825080792778, 8.925382904982964431304135137785, 10.49613070020522887822505272928, 11.36268702034390046216762887161
