L(s) = 1 | + (171. + 1.00e3i)2-s − 3.40e4i·3-s + (−9.89e5 + 3.46e5i)4-s + 1.74e7·5-s + (3.44e7 − 5.85e6i)6-s − 3.10e8i·7-s + (−5.19e8 − 9.39e8i)8-s − 1.16e9·9-s + (2.99e9 + 1.76e10i)10-s − 2.01e9i·11-s + (1.18e10 + 3.37e10i)12-s − 1.45e11·13-s + (3.13e11 − 5.33e10i)14-s − 5.94e11i·15-s + (8.59e11 − 6.86e11i)16-s − 3.62e12·17-s + ⋯ |
L(s) = 1 | + (0.167 + 0.985i)2-s − 0.577i·3-s + (−0.943 + 0.330i)4-s + 1.78·5-s + (0.569 − 0.0968i)6-s − 1.10i·7-s + (−0.484 − 0.874i)8-s − 0.333·9-s + (0.299 + 1.76i)10-s − 0.0776i·11-s + (0.190 + 0.544i)12-s − 1.05·13-s + (1.08 − 0.184i)14-s − 1.03i·15-s + (0.781 − 0.624i)16-s − 1.80·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{21}{2})\) |
\(\approx\) |
\(1.688474656\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.688474656\) |
\(L(11)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-171. - 1.00e3i)T \) |
| 3 | \( 1 + 3.40e4iT \) |
good | 5 | \( 1 - 1.74e7T + 9.53e13T^{2} \) |
| 7 | \( 1 + 3.10e8iT - 7.97e16T^{2} \) |
| 11 | \( 1 + 2.01e9iT - 6.72e20T^{2} \) |
| 13 | \( 1 + 1.45e11T + 1.90e22T^{2} \) |
| 17 | \( 1 + 3.62e12T + 4.06e24T^{2} \) |
| 19 | \( 1 - 3.83e11iT - 3.75e25T^{2} \) |
| 23 | \( 1 + 7.19e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 + 2.72e14T + 1.76e29T^{2} \) |
| 31 | \( 1 + 3.17e14iT - 6.71e29T^{2} \) |
| 37 | \( 1 - 1.82e15T + 2.31e31T^{2} \) |
| 41 | \( 1 - 7.50e15T + 1.80e32T^{2} \) |
| 43 | \( 1 + 9.17e15iT - 4.67e32T^{2} \) |
| 47 | \( 1 + 7.76e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 + 5.62e16T + 3.05e34T^{2} \) |
| 59 | \( 1 - 3.58e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 + 6.85e17T + 5.08e35T^{2} \) |
| 67 | \( 1 + 1.47e18iT - 3.32e36T^{2} \) |
| 71 | \( 1 - 5.13e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 8.06e17T + 1.84e37T^{2} \) |
| 79 | \( 1 + 1.03e19iT - 8.96e37T^{2} \) |
| 83 | \( 1 + 2.51e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 + 2.22e18T + 9.72e38T^{2} \) |
| 97 | \( 1 - 5.14e19T + 5.43e39T^{2} \) |
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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
−14.63175056565729632689299229747, −13.65648097040950141576165051990, −12.91404734224499768391121158937, −10.31620986670197177809108828278, −8.966491440808276245205118487835, −7.12958420967936732271155495270, −6.18542695030811600606496966432, −4.66273491035336726005534266111, −2.25396181081471286419438952962, −0.46136692795183301335723454395,
1.83602366771352615704004110225, 2.69260237241838797429041187428, 4.86902870138973751942562124107, 5.90195331073157242511498752816, 9.126364629273776941157112887367, 9.648918855250819221512928804090, 11.12651507631305010175797190181, 12.74223873563335195184343115354, 13.89675786506340656494360295719, 15.18039416499978401178303235817