| L(s) = 1 | + (−25.8 + 18.8i)2-s + (80.7 + 248. i)3-s + (316. − 973. i)4-s + (7.36e3 + 5.35e3i)5-s + (−6.76e3 − 4.91e3i)6-s + (1.13e4 − 3.47e4i)7-s + (1.01e4 + 3.11e4i)8-s + (8.80e4 − 6.39e4i)9-s − 2.91e5·10-s + (−1.99e5 − 4.95e5i)11-s + 2.67e5·12-s + (1.44e6 − 1.05e6i)13-s + (3.61e5 + 1.11e6i)14-s + (−7.35e5 + 2.26e6i)15-s + (−8.48e5 − 6.16e5i)16-s + (−2.38e6 − 1.73e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.191 + 0.590i)3-s + (0.154 − 0.475i)4-s + (1.05 + 0.766i)5-s + (−0.355 − 0.258i)6-s + (0.254 − 0.782i)7-s + (0.109 + 0.336i)8-s + (0.496 − 0.361i)9-s − 0.921·10-s + (−0.373 − 0.927i)11-s + 0.310·12-s + (1.07 − 0.784i)13-s + (0.179 + 0.553i)14-s + (−0.250 + 0.769i)15-s + (−0.202 − 0.146i)16-s + (−0.407 − 0.296i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.88685 + 0.660617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88685 + 0.660617i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (25.8 - 18.8i)T \) |
| 11 | \( 1 + (1.99e5 + 4.95e5i)T \) |
| good | 3 | \( 1 + (-80.7 - 248. i)T + (-1.43e5 + 1.04e5i)T^{2} \) |
| 5 | \( 1 + (-7.36e3 - 5.35e3i)T + (1.50e7 + 4.64e7i)T^{2} \) |
| 7 | \( 1 + (-1.13e4 + 3.47e4i)T + (-1.59e9 - 1.16e9i)T^{2} \) |
| 13 | \( 1 + (-1.44e6 + 1.05e6i)T + (5.53e11 - 1.70e12i)T^{2} \) |
| 17 | \( 1 + (2.38e6 + 1.73e6i)T + (1.05e13 + 3.25e13i)T^{2} \) |
| 19 | \( 1 + (-4.96e6 - 1.52e7i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 - 5.59e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (1.03e6 - 3.18e6i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (1.47e8 - 1.07e8i)T + (7.85e15 - 2.41e16i)T^{2} \) |
| 37 | \( 1 + (6.08e7 - 1.87e8i)T + (-1.43e17 - 1.04e17i)T^{2} \) |
| 41 | \( 1 + (4.51e8 + 1.39e9i)T + (-4.45e17 + 3.23e17i)T^{2} \) |
| 43 | \( 1 + 8.77e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-3.14e8 - 9.68e8i)T + (-2.00e18 + 1.45e18i)T^{2} \) |
| 53 | \( 1 + (-1.78e9 + 1.29e9i)T + (2.86e18 - 8.81e18i)T^{2} \) |
| 59 | \( 1 + (1.95e9 - 6.00e9i)T + (-2.43e19 - 1.77e19i)T^{2} \) |
| 61 | \( 1 + (-5.82e9 - 4.23e9i)T + (1.34e19 + 4.13e19i)T^{2} \) |
| 67 | \( 1 - 4.22e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (4.17e9 + 3.03e9i)T + (7.14e19 + 2.19e20i)T^{2} \) |
| 73 | \( 1 + (1.87e9 - 5.75e9i)T + (-2.53e20 - 1.84e20i)T^{2} \) |
| 79 | \( 1 + (-2.20e10 + 1.60e10i)T + (2.31e20 - 7.11e20i)T^{2} \) |
| 83 | \( 1 + (3.40e10 + 2.47e10i)T + (3.97e20 + 1.22e21i)T^{2} \) |
| 89 | \( 1 + 9.05e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-2.51e10 + 1.82e10i)T + (2.21e21 - 6.80e21i)T^{2} \) |
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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
−15.60004863119547552498201590083, −14.37353588528447302899378418750, −13.36087361654956778577094347935, −10.79821197801177134498470912986, −10.22655822939889266414509703904, −8.775404694385174909704342582724, −7.04281403119982200984610063994, −5.61176549935974095600918417140, −3.36419896245578008505419641071, −1.16523400538057391913250387490,
1.32356879248000567290583440251, 2.25008899076658485154255060474, 4.97867732001996206496695858279, 6.89783472792831467139976587954, 8.631942098716803891855050538330, 9.549110696348895179236713923088, 11.24145893866563199137744957014, 12.85825700881835242642312019903, 13.39246331722137442731799447947, 15.38371009716157551354572018119
