L(s) = 1 | + (0.447 − 0.774i)2-s + (1.22 + 1.22i)3-s + (0.599 + 1.03i)4-s + (−1.87 − 3.24i)5-s + (1.49 − 0.401i)6-s + (−0.725 + 1.25i)7-s + 2.86·8-s + (0.00384 + 2.99i)9-s − 3.35·10-s + (−0.5 + 0.866i)11-s + (−0.536 + 2.00i)12-s + (−2.87 − 4.98i)13-s + (0.648 + 1.12i)14-s + (1.67 − 6.27i)15-s + (0.0800 − 0.138i)16-s − 4.79·17-s + ⋯ |
L(s) = 1 | + (0.316 − 0.547i)2-s + (0.707 + 0.706i)3-s + (0.299 + 0.519i)4-s + (−0.838 − 1.45i)5-s + (0.610 − 0.164i)6-s + (−0.274 + 0.474i)7-s + 1.01·8-s + (0.00128 + 0.999i)9-s − 1.06·10-s + (−0.150 + 0.261i)11-s + (−0.154 + 0.579i)12-s + (−0.798 − 1.38i)13-s + (0.173 + 0.300i)14-s + (0.432 − 1.61i)15-s + (0.0200 − 0.0346i)16-s − 1.16·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31630 - 0.114312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31630 - 0.114312i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.447 + 0.774i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.87 + 3.24i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.725 - 1.25i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (2.87 + 4.98i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 0.702T + 19T^{2} \) |
| 23 | \( 1 + (-0.825 - 1.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.15 + 3.72i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.65 - 2.86i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 + (2.12 + 3.67i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.05 + 3.55i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.898 - 1.55i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 + (-2.32 - 4.02i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.27 - 2.20i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.47 + 7.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.14T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + (-0.543 + 0.941i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.90 + 3.29i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 + (1.64 - 2.85i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.48744339972181298404034992029, −12.74395279961215987922752073318, −12.00881297090682081789954659508, −10.78875164902541139869842643641, −9.463464412505869606654904543640, −8.374108585475706663460734330347, −7.63944406405573174015519526584, −5.07094737039958634117753956355, −4.13303628190339915877173735442, −2.68795680624850277873737133833,
2.53264539477607638353543442603, 4.20396939852830988930132799869, 6.58999612750236329830392779345, 6.87426322366561269074669448843, 7.912919709698364022535120311952, 9.612619986259039467935072649630, 10.89851348719835940624499416111, 11.71717730941117642097035276205, 13.30563508595311377656147436803, 14.22609111974396112909102139623