L(s) = 1 | − 3-s − 1.36·5-s − 7-s + 9-s − 5.49·11-s + 6.48·13-s + 1.36·15-s + 7.12·17-s + 3.85·19-s + 21-s + 23-s − 3.13·25-s − 27-s − 6.11·29-s + 5.12·31-s + 5.49·33-s + 1.36·35-s + 6.11·37-s − 6.48·39-s + 6.11·41-s − 10.1·43-s − 1.36·45-s + 1.00·47-s + 49-s − 7.12·51-s − 9.70·53-s + 7.51·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.610·5-s − 0.377·7-s + 0.333·9-s − 1.65·11-s + 1.79·13-s + 0.352·15-s + 1.72·17-s + 0.883·19-s + 0.218·21-s + 0.208·23-s − 0.626·25-s − 0.192·27-s − 1.13·29-s + 0.919·31-s + 0.957·33-s + 0.230·35-s + 1.00·37-s − 1.03·39-s + 0.954·41-s − 1.54·43-s − 0.203·45-s + 0.147·47-s + 0.142·49-s − 0.997·51-s − 1.33·53-s + 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.222370236\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.222370236\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 1.36T + 5T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 29 | \( 1 + 6.11T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 - 6.11T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.00T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 + 0.754T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 + 1.97T + 67T^{2} \) |
| 71 | \( 1 + 6.20T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 - 5.49T + 79T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79300416263993892496749312811, −7.39928349542061477586015264465, −6.26254935103106955795790687928, −5.78724992845178542693617825152, −5.22514149932849615997625082874, −4.30642653584781865656582889669, −3.41451130176187605649505031326, −2.99346176723810677507822285932, −1.54695494321202444580371487863, −0.59457269870187268847450115063,
0.59457269870187268847450115063, 1.54695494321202444580371487863, 2.99346176723810677507822285932, 3.41451130176187605649505031326, 4.30642653584781865656582889669, 5.22514149932849615997625082874, 5.78724992845178542693617825152, 6.26254935103106955795790687928, 7.39928349542061477586015264465, 7.79300416263993892496749312811