L(s) = 1 | + (−723. + 17.4i)2-s + (2.86e3 + 3.39e4i)3-s + (5.23e5 − 2.52e4i)4-s + 5.49e6i·5-s + (−2.66e6 − 2.45e7i)6-s − 1.05e8i·7-s + (−3.78e8 + 2.74e7i)8-s + (−1.14e9 + 1.94e8i)9-s + (−9.57e7 − 3.97e9i)10-s + 7.09e9·11-s + (2.35e9 + 1.77e10i)12-s − 6.63e10·13-s + (1.83e9 + 7.61e10i)14-s + (−1.86e11 + 1.57e10i)15-s + (2.73e11 − 2.64e10i)16-s + 6.50e11i·17-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0240i)2-s + (0.0840 + 0.996i)3-s + (0.998 − 0.0481i)4-s + 1.25i·5-s + (−0.108 − 0.994i)6-s − 0.985i·7-s + (−0.997 + 0.0721i)8-s + (−0.985 + 0.167i)9-s + (−0.0302 − 1.25i)10-s + 0.907·11-s + (0.131 + 0.991i)12-s − 1.73·13-s + (0.0237 + 0.985i)14-s + (−1.25 + 0.105i)15-s + (0.995 − 0.0961i)16-s + 1.33i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(0.1690455729\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1690455729\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (723. - 17.4i)T \) |
| 3 | \( 1 + (-2.86e3 - 3.39e4i)T \) |
good | 5 | \( 1 - 5.49e6iT - 1.90e13T^{2} \) |
| 7 | \( 1 + 1.05e8iT - 1.13e16T^{2} \) |
| 11 | \( 1 - 7.09e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 6.63e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 6.50e11iT - 2.39e23T^{2} \) |
| 19 | \( 1 + 8.23e11iT - 1.97e24T^{2} \) |
| 23 | \( 1 + 1.08e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 9.35e13iT - 6.10e27T^{2} \) |
| 31 | \( 1 + 1.40e14iT - 2.16e28T^{2} \) |
| 37 | \( 1 - 2.24e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 3.53e14iT - 4.39e30T^{2} \) |
| 43 | \( 1 - 1.91e15iT - 1.08e31T^{2} \) |
| 47 | \( 1 - 1.54e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.30e16iT - 5.77e32T^{2} \) |
| 59 | \( 1 - 7.28e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 3.87e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.26e17iT - 4.95e34T^{2} \) |
| 71 | \( 1 + 1.37e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 2.84e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 3.04e17iT - 1.13e36T^{2} \) |
| 83 | \( 1 + 1.54e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 1.94e18iT - 1.09e37T^{2} \) |
| 97 | \( 1 - 3.32e17T + 5.60e37T^{2} \) |
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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
−15.14920193690793439206193080000, −14.40343252499564372877020756792, −11.59682166291954148383320819804, −10.42557134970600726022869057269, −9.682930978087865769451091747452, −7.76027325828901455805049792873, −6.39119637435027347005835796564, −3.92269198058917268255338242287, −2.39178673385113371484106937126, −0.07687696174522113423485751041,
1.24739192544902601804504870981, 2.47287806496339004549303576393, 5.45372497626263961740098269437, 7.14049967317676511227302466026, 8.542684490758012939538316138169, 9.457019938516775470184874927840, 12.03379185275537976136440420971, 12.26414758066082181475443082506, 14.45990601252013401312445931827, 16.23920461983957916501467085202