L(s) = 1 | + (0.780 + 1.17i)2-s − 3.02·3-s + (−0.780 + 1.84i)4-s + (−2.35 − 3.56i)6-s + (2.17 + 1.51i)7-s + (−2.78 + 0.516i)8-s + 6.12·9-s + 4.71i·11-s + (2.35 − 5.56i)12-s + 2i·13-s + (−0.0846 + 3.74i)14-s + (−2.78 − 2.87i)16-s + 1.12i·17-s + (4.78 + 7.22i)18-s − 4.71·19-s + ⋯ |
L(s) = 1 | + (0.552 + 0.833i)2-s − 1.74·3-s + (−0.390 + 0.920i)4-s + (−0.962 − 1.45i)6-s + (0.821 + 0.570i)7-s + (−0.983 + 0.182i)8-s + 2.04·9-s + 1.42i·11-s + (0.680 − 1.60i)12-s + 0.554i·13-s + (−0.0226 + 0.999i)14-s + (−0.695 − 0.718i)16-s + 0.272i·17-s + (1.12 + 1.70i)18-s − 1.08·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.182249 - 0.631116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.182249 - 0.631116i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.780 - 1.17i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.17 - 1.51i)T \) |
good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 11 | \( 1 - 4.71iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 1.12iT - 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 + 6.41iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.12iT - 41T^{2} \) |
| 43 | \( 1 - 0.371iT - 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 2.06T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 3.76iT - 67T^{2} \) |
| 71 | \( 1 + 7.36iT - 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 1.32iT - 79T^{2} \) |
| 83 | \( 1 + 3.02T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 1.12iT - 97T^{2} \) |
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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
−11.17503114184919234578918213230, −10.29440605715213328049816290155, −9.170045227758913860301602957720, −8.103267878192805863244453711880, −7.04550254296000320339428749470, −6.46586060993693410299772898504, −5.57572153534919815951305493067, −4.72065974699232172361418167409, −4.28271977834221842183138578843, −2.01250908318250598128411808139,
0.36761310851017820312551821314, 1.56271826070184002219109869152, 3.49066811595147539940116240328, 4.51851568417404845399739315815, 5.41501156752258952979253614170, 5.91215190704036765044015943724, 6.96525343715574119153586700773, 8.221816712469989244560844824283, 9.480124217989586299097375044635, 10.60887362605522884573084067359