L(s) = 1 | + (723. + 17.4i)2-s + (−2.86e3 − 3.39e4i)3-s + (5.23e5 + 2.52e4i)4-s + 5.49e6i·5-s + (−1.48e6 − 2.46e7i)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 + (−6.43e8 − 1.78e10i)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 + (−0.0600 − 0.998i)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 + (−0.0359 − 0.999i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(2.511793317\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.511793317\) |
\(L(\frac{21}{2})\) |
|
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
−15.18127939745174999591918879147, −14.49069574167572928254882692523, −12.93013170541104013177315196612, −11.94757367999841011542038499797, −10.55961924000322895225127515283, −7.80144387217403824138449101198, −6.62940910757103460984632056394, −5.38308800567532671085685450568, −2.91876457177305249631246779087, −2.14629629828158637412255930112,
0.53829853040758446945923904512, 2.84640465004266078888375689197, 4.67369516887414929662770744336, 5.05923229654120571178921634779, 7.42666816342303112925832226979, 9.502454265134908711528094766240, 10.92580688451190492750771804594, 12.42732532320741410637931783499, 13.70347583491791151750332740495, 15.11509165903866829357980687873