| L(s) = 1 | + 3.09i·3-s − 0.646i·5-s + (2.44 + i)7-s − 6.58·9-s + (−2.79 + 1.79i)11-s + 3.09·13-s + 2·15-s − 3.74·17-s − 5.54·19-s + (−3.09 + 7.58i)21-s − 4·23-s + 4.58·25-s − 11.0i·27-s + 7.58i·29-s + 1.15i·31-s + ⋯ |
| L(s) = 1 | + 1.78i·3-s − 0.288i·5-s + (0.925 + 0.377i)7-s − 2.19·9-s + (−0.841 + 0.540i)11-s + 0.858·13-s + 0.516·15-s − 0.907·17-s − 1.27·19-s + (−0.675 + 1.65i)21-s − 0.834·23-s + 0.916·25-s − 2.13i·27-s + 1.40i·29-s + 0.207i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.095869007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095869007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.44 - i)T \) |
| 11 | \( 1 + (2.79 - 1.79i)T \) |
| good | 3 | \( 1 - 3.09iT - 3T^{2} \) |
| 5 | \( 1 + 0.646iT - 5T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.58iT - 29T^{2} \) |
| 31 | \( 1 - 1.15iT - 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 5.03iT - 47T^{2} \) |
| 53 | \( 1 + 2.41T + 53T^{2} \) |
| 59 | \( 1 + 3.09iT - 59T^{2} \) |
| 61 | \( 1 - 9.28T + 61T^{2} \) |
| 67 | \( 1 + 1.58T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−10.30179680191697449190704797040, −9.295668027046030646890089721858, −8.561888158913981340464276027516, −8.243872517327268788914428964528, −6.70302447815142628148302354155, −5.57311722131099583724090444199, −4.85463283922883990047114768042, −4.36522403144546032679745119589, −3.28432287398258393339868114816, −2.03242403495145099897177013641,
0.43863228130199835588013601423, 1.80028368214641635297925343160, 2.52408310799388441051607437595, 3.95500424272174485585359917857, 5.27563583878506862934448692934, 6.22151746336300701961672289583, 6.78608633758288269492202885365, 7.69941166224904500735505724129, 8.320225894367046678319763116459, 8.753875152826639183497623008137
