| L(s) = 1 | + (16 + 27.7i)2-s + (121.5 − 210. i)3-s + (−511. + 886. i)4-s + (−1.63e3 − 2.84e3i)5-s + 7.77e3·6-s + (−1.70e4 + 4.10e4i)7-s − 3.27e4·8-s + (−2.95e4 − 5.11e4i)9-s + (5.24e4 − 9.08e4i)10-s + (1.01e5 − 1.76e5i)11-s + (1.24e5 + 2.15e5i)12-s + 1.41e6·13-s + (−1.41e6 + 1.83e5i)14-s − 7.97e5·15-s + (−5.24e5 − 9.08e5i)16-s + (−8.69e5 + 1.50e6i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.234 − 0.406i)5-s + 0.408·6-s + (−0.384 + 0.923i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.165 − 0.287i)10-s + (0.190 − 0.330i)11-s + (0.144 + 0.249i)12-s + 1.05·13-s + (−0.701 + 0.0911i)14-s − 0.271·15-s + (−0.125 − 0.216i)16-s + (−0.148 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0658 + 0.997i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.0658 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.01290 - 0.948291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01290 - 0.948291i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 - 27.7i)T \) |
| 3 | \( 1 + (-121.5 + 210. i)T \) |
| 7 | \( 1 + (1.70e4 - 4.10e4i)T \) |
| good | 5 | \( 1 + (1.63e3 + 2.84e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-1.01e5 + 1.76e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.41e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (8.69e5 - 1.50e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (9.98e6 + 1.73e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.79e7 + 3.10e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.92e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-2.12e6 + 3.67e6i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (2.49e8 + 4.31e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 7.24e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.43e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.11e9 + 1.93e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.23e9 + 2.14e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.11e7 + 1.93e7i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.07e8 - 3.59e8i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (3.95e9 - 6.85e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.48e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (1.23e10 - 2.13e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-6.25e9 - 1.08e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 5.94e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.91e10 - 5.05e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 3.15e10T + 7.15e21T^{2} \) |
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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
−13.20308817156451690032559667427, −12.53934502210504587726114071038, −11.16990669609829376238424721520, −9.023452578627253561897524577431, −8.412006073772018271931009525215, −6.77721124557230683524840089659, −5.71971195152157850661582797710, −4.01685612295515556389904437200, −2.38722564863870628377707578942, −0.35948957950941894758679905946,
1.54328502155598745667994308900, 3.39939556134343824058799217949, 4.14374056109600764419101737621, 6.01228644873917225012091434357, 7.64970727755899011658798001806, 9.299619781925671046696677829642, 10.43042039993236618836085783189, 11.25954870967564705542445352747, 12.78800823870098196004926295743, 13.84670524812118910899237921370
