| 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
−13.84670524812118910899237921370, −12.78800823870098196004926295743, −11.25954870967564705542445352747, −10.43042039993236618836085783189, −9.299619781925671046696677829642, −7.64970727755899011658798001806, −6.01228644873917225012091434357, −4.14374056109600764419101737621, −3.39939556134343824058799217949, −1.54328502155598745667994308900,
0.35948957950941894758679905946, 2.38722564863870628377707578942, 4.01685612295515556389904437200, 5.71971195152157850661582797710, 6.77721124557230683524840089659, 8.412006073772018271931009525215, 9.023452578627253561897524577431, 11.16990669609829376238424721520, 12.53934502210504587726114071038, 13.20308817156451690032559667427
