L(s) = 1 | + 2.23·3-s − 2.23i·5-s − 2.64i·7-s + 2.00·9-s + 5.91·11-s + 6.70i·13-s − 5.00i·15-s + 7.93·17-s − 5.91i·21-s − 5.00·25-s − 2.23·27-s − 5.91i·29-s + 13.2·33-s − 5.91·35-s + 15.0i·39-s + ⋯ |
L(s) = 1 | + 1.29·3-s − 0.999i·5-s − 0.999i·7-s + 0.666·9-s + 1.78·11-s + 1.86i·13-s − 1.29i·15-s + 1.92·17-s − 1.29i·21-s − 1.00·25-s − 0.430·27-s − 1.09i·29-s + 2.30·33-s − 0.999·35-s + 2.40i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.247221378\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.247221378\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 6.70iT - 13T^{2} \) |
| 17 | \( 1 - 7.93T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 7.93iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - iT - 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−9.106530822878168836460859036550, −8.202229681275867930657656261525, −7.60610344126290992685876276359, −6.77245225815496850241367131586, −5.88403378513276906543247334333, −4.47594567930187953495049302687, −4.05922632874805162625118141169, −3.33534818490040472104395224826, −1.81767919423031366124221398371, −1.15737014343272202687890176959,
1.40633029084067118278971789770, 2.62529142033039355000664457893, 3.28700543291341579496130000792, 3.72063509885539824016954138762, 5.36875627571746295019514751953, 5.96785716132995411517653404715, 6.97001730294721334992375775919, 7.74952236711717523753894697047, 8.363304995316357553177855217852, 9.054586497814248133148861962068