L(s) = 1 | + (1.72 − 2.24i)2-s + (−2.05 − 7.73i)4-s + 6.58i·5-s + (15.1 + 10.5i)7-s + (−20.8 − 8.73i)8-s + (14.7 + 11.3i)10-s − 54.2i·11-s − 40.9i·13-s + (49.9 − 15.7i)14-s + (−55.5 + 31.7i)16-s − 69.4i·17-s + 160.·19-s + (50.9 − 13.5i)20-s + (−121. − 93.5i)22-s − 87.8i·23-s + ⋯ |
L(s) = 1 | + (0.609 − 0.792i)2-s + (−0.256 − 0.966i)4-s + 0.589i·5-s + (0.820 + 0.571i)7-s + (−0.922 − 0.386i)8-s + (0.467 + 0.359i)10-s − 1.48i·11-s − 0.874i·13-s + (0.953 − 0.301i)14-s + (−0.868 + 0.495i)16-s − 0.991i·17-s + 1.93·19-s + (0.569 − 0.151i)20-s + (−1.17 − 0.906i)22-s − 0.796i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.437836730\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.437836730\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.72 + 2.24i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-15.1 - 10.5i)T \) |
good | 5 | \( 1 - 6.58iT - 125T^{2} \) |
| 11 | \( 1 + 54.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 40.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 69.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 160.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 87.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 112. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 194. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 267. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 275. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 270. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.23e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 691. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 381. iT - 9.12e5T^{2} \) |
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
−11.28527226103940059943418487661, −10.77307316953761748196738421612, −9.545772636956611077805105039313, −8.569010632474931657221454104727, −7.27049946801590237656459022178, −5.73878615331713265116264980870, −5.20035958864903271405114480800, −3.49089054673366862182928767667, −2.62145203413580239635247992802, −0.855297038793582759731001468808,
1.68883083664318660673702655035, 3.78589827993522940173882319154, 4.71434450008683281409381530625, 5.57405268269448799332934946170, 7.17575229998023493699315393463, 7.56866404378206579119236738352, 8.856511107230986593062076680849, 9.729335160487656257229842228472, 11.23797871550936265256080423921, 12.08621067712257037439483724705