L(s) = 1 | + 3.16i·5-s + 7-s − 5.44i·11-s − 3.61i·13-s − 3.27·17-s + 3.20i·19-s + 0.673·23-s − 5.04·25-s − 2.85i·29-s − 3.71·31-s + 3.16i·35-s + 11.9i·37-s + 7.44·41-s + 12.5i·43-s − 4.06·47-s + ⋯ |
L(s) = 1 | + 1.41i·5-s + 0.377·7-s − 1.64i·11-s − 1.00i·13-s − 0.794·17-s + 0.735i·19-s + 0.140·23-s − 1.00·25-s − 0.529i·29-s − 0.667·31-s + 0.535i·35-s + 1.97i·37-s + 1.16·41-s + 1.91i·43-s − 0.592·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.808547831\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.808547831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3.16iT - 5T^{2} \) |
| 11 | \( 1 + 5.44iT - 11T^{2} \) |
| 13 | \( 1 + 3.61iT - 13T^{2} \) |
| 17 | \( 1 + 3.27T + 17T^{2} \) |
| 19 | \( 1 - 3.20iT - 19T^{2} \) |
| 23 | \( 1 - 0.673T + 23T^{2} \) |
| 29 | \( 1 + 2.85iT - 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 11.9iT - 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 - 12.5iT - 43T^{2} \) |
| 47 | \( 1 + 4.06T + 47T^{2} \) |
| 53 | \( 1 + 0.291iT - 53T^{2} \) |
| 59 | \( 1 - 0.0209iT - 59T^{2} \) |
| 61 | \( 1 - 5.34iT - 61T^{2} \) |
| 67 | \( 1 + 6.20iT - 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 1.35T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.6iT - 83T^{2} \) |
| 89 | \( 1 - 5.93T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−7.971864268452896617811147667019, −7.70202896696784233337290265970, −6.51844034314124381650515270641, −6.26157250873564357787569723177, −5.49531739439025032217401313998, −4.53913180348086213302932622103, −3.40455554822442508712952945862, −3.12792172772313041233700719867, −2.18905236669551308591336683114, −0.857668939277944377439496754356,
0.57909269510595634065377613383, 1.85158276751386797058263131351, 2.19390002763051045746961623566, 3.86999900917113686739466322528, 4.41279009639881240258778434804, 4.97888947438283762128096712176, 5.56685865457164484222673170023, 6.79122838902687277370336040254, 7.16809607474331218507460775008, 7.998145923576016018688469265306