L(s) = 1 | + (477. + 185. i)2-s + 2.70e4i·3-s + (1.93e5 + 1.77e5i)4-s − 2.11e6·5-s + (−5.03e6 + 1.29e7i)6-s − 3.35e7i·7-s + (5.92e7 + 1.20e8i)8-s − 3.46e8·9-s + (−1.01e9 − 3.93e8i)10-s + 3.87e9i·11-s + (−4.80e9 + 5.23e9i)12-s + 7.06e8·13-s + (6.23e9 − 1.60e10i)14-s − 5.74e10i·15-s + (5.90e9 + 6.84e10i)16-s + 2.14e11·17-s + ⋯ |
L(s) = 1 | + (0.931 + 0.362i)2-s + 1.37i·3-s + (0.736 + 0.676i)4-s − 1.08·5-s + (−0.499 + 1.28i)6-s − 0.831i·7-s + (0.441 + 0.897i)8-s − 0.894·9-s + (−1.01 − 0.393i)10-s + 1.64i·11-s + (−0.930 + 1.01i)12-s + 0.0665·13-s + (0.301 − 0.774i)14-s − 1.49i·15-s + (0.0858 + 0.996i)16-s + 1.81·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(0.859268 + 2.20749i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.859268 + 2.20749i\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-477. - 185. i)T \) |
good | 3 | \( 1 - 2.70e4iT - 3.87e8T^{2} \) |
| 5 | \( 1 + 2.11e6T + 3.81e12T^{2} \) |
| 7 | \( 1 + 3.35e7iT - 1.62e15T^{2} \) |
| 11 | \( 1 - 3.87e9iT - 5.55e18T^{2} \) |
| 13 | \( 1 - 7.06e8T + 1.12e20T^{2} \) |
| 17 | \( 1 - 2.14e11T + 1.40e22T^{2} \) |
| 19 | \( 1 - 5.73e9iT - 1.04e23T^{2} \) |
| 23 | \( 1 + 1.10e12iT - 3.24e24T^{2} \) |
| 29 | \( 1 - 1.21e13T + 2.10e26T^{2} \) |
| 31 | \( 1 + 5.37e12iT - 6.99e26T^{2} \) |
| 37 | \( 1 + 2.74e13T + 1.68e28T^{2} \) |
| 41 | \( 1 - 2.53e14T + 1.07e29T^{2} \) |
| 43 | \( 1 + 3.34e14iT - 2.52e29T^{2} \) |
| 47 | \( 1 + 1.35e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 + 3.47e15T + 1.08e31T^{2} \) |
| 59 | \( 1 - 3.11e15iT - 7.50e31T^{2} \) |
| 61 | \( 1 + 6.55e15T + 1.36e32T^{2} \) |
| 67 | \( 1 - 8.23e15iT - 7.40e32T^{2} \) |
| 71 | \( 1 + 7.54e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 9.58e15T + 3.46e33T^{2} \) |
| 79 | \( 1 - 1.19e17iT - 1.43e34T^{2} \) |
| 83 | \( 1 + 1.86e17iT - 3.49e34T^{2} \) |
| 89 | \( 1 - 1.85e17T + 1.22e35T^{2} \) |
| 97 | \( 1 - 7.14e17T + 5.77e35T^{2} \) |
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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
−21.02893579015721028308937053921, −20.07112671489246898832667157659, −16.83009800257283918117799087239, −15.64687175239783165034213087019, −14.57985366953404570191153937811, −12.16103381134670891384900938670, −10.32650744788821474012740838065, −7.52432762455662688913262090481, −4.70403803305665355441455857565, −3.67606704564291121445981407910,
0.994155710131953499849212578784, 3.14026800099141063940624539318, 5.92743960769429963686139471082, 7.85408510370928911101879306299, 11.51600816648073989413452280196, 12.48731644909282166916345360931, 14.09911598901124001459567102370, 15.95707541059625132977699214521, 18.82194797828805241595354965278, 19.35914563053143045926712953981