| L(s) = 1 | + (−8.67 − 26.6i)2-s + (588. + 427. i)3-s + (1.01e3 − 740. i)4-s + (−3.28e3 + 1.01e4i)5-s + (6.31e3 − 1.94e4i)6-s + (−3.32e4 + 2.41e4i)7-s + (−7.51e4 − 5.45e4i)8-s + (1.09e5 + 3.35e5i)9-s + 2.98e5·10-s + (5.34e5 + 1.80e3i)11-s + 9.17e5·12-s + (1.72e5 + 5.29e5i)13-s + (9.34e5 + 6.78e5i)14-s + (−6.26e6 + 4.55e6i)15-s + (−7.46e3 + 2.29e4i)16-s + (2.07e6 − 6.38e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.191 − 0.589i)2-s + (1.39 + 1.01i)3-s + (0.497 − 0.361i)4-s + (−0.470 + 1.44i)5-s + (0.331 − 1.02i)6-s + (−0.748 + 0.544i)7-s + (−0.810 − 0.588i)8-s + (0.615 + 1.89i)9-s + 0.943·10-s + (0.999 + 0.00337i)11-s + 1.06·12-s + (0.128 + 0.395i)13-s + (0.464 + 0.337i)14-s + (−2.12 + 1.54i)15-s + (−0.00177 + 0.00547i)16-s + (0.354 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.11115 + 1.01115i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11115 + 1.01115i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-5.34e5 - 1.80e3i)T \) |
| good | 2 | \( 1 + (8.67 + 26.6i)T + (-1.65e3 + 1.20e3i)T^{2} \) |
| 3 | \( 1 + (-588. - 427. i)T + (5.47e4 + 1.68e5i)T^{2} \) |
| 5 | \( 1 + (3.28e3 - 1.01e4i)T + (-3.95e7 - 2.87e7i)T^{2} \) |
| 7 | \( 1 + (3.32e4 - 2.41e4i)T + (6.11e8 - 1.88e9i)T^{2} \) |
| 13 | \( 1 + (-1.72e5 - 5.29e5i)T + (-1.44e12 + 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-2.07e6 + 6.38e6i)T + (-2.77e13 - 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-3.61e5 - 2.62e5i)T + (3.59e13 + 1.10e14i)T^{2} \) |
| 23 | \( 1 - 3.42e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-9.60e7 + 6.97e7i)T + (3.77e15 - 1.16e16i)T^{2} \) |
| 31 | \( 1 + (1.18e7 + 3.65e7i)T + (-2.05e16 + 1.49e16i)T^{2} \) |
| 37 | \( 1 + (1.95e8 - 1.42e8i)T + (5.49e16 - 1.69e17i)T^{2} \) |
| 41 | \( 1 + (9.42e8 + 6.85e8i)T + (1.70e17 + 5.23e17i)T^{2} \) |
| 43 | \( 1 - 1.32e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.58e9 + 1.15e9i)T + (7.63e17 + 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-3.75e8 - 1.15e9i)T + (-7.49e18 + 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-5.08e9 + 3.69e9i)T + (9.31e18 - 2.86e19i)T^{2} \) |
| 61 | \( 1 + (9.43e8 - 2.90e9i)T + (-3.52e19 - 2.55e19i)T^{2} \) |
| 67 | \( 1 - 7.26e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (1.18e9 - 3.65e9i)T + (-1.86e20 - 1.35e20i)T^{2} \) |
| 73 | \( 1 + (1.43e10 - 1.03e10i)T + (9.69e19 - 2.98e20i)T^{2} \) |
| 79 | \( 1 + (1.44e9 + 4.46e9i)T + (-6.05e20 + 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-3.36e9 + 1.03e10i)T + (-1.04e21 - 7.56e20i)T^{2} \) |
| 89 | \( 1 - 6.62e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (2.38e10 + 7.32e10i)T + (-5.78e21 + 4.20e21i)T^{2} \) |
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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
−18.92468896897901988576509061785, −15.93143358202429064157188836242, −15.09700799042701953968376345879, −14.16124493327187430692251636345, −11.59719050160632902878123309317, −10.20165386188973854751634761355, −9.169779887681391357394417326358, −6.87320545272836789794786084838, −3.47709033955139161154529716441, −2.60796005196503949563929265095,
1.21691091520118598713241421766, 3.44899925305696544915214669467, 6.75269120563742761804105817655, 8.102583923952440597612240584103, 8.983730844714617227042371828191, 12.28123831188792525870873620412, 13.09584315926782543144304933360, 14.76447159263259780005941521828, 16.20769238875456137791523780750, 17.33938193426900379985681679844
