L(s) = 1 | + i·3-s + (−0.311 + 2.21i)5-s − i·7-s − 9-s + 5.05·11-s + 3.37i·13-s + (−2.21 − 0.311i)15-s − 7.18i·17-s + 8.23·19-s + 21-s + 6.23i·23-s + (−4.80 − 1.37i)25-s − i·27-s + 2·29-s + 4.62·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.139 + 0.990i)5-s − 0.377i·7-s − 0.333·9-s + 1.52·11-s + 0.936i·13-s + (−0.571 − 0.0803i)15-s − 1.74i·17-s + 1.88·19-s + 0.218·21-s + 1.30i·23-s + (−0.961 − 0.275i)25-s − 0.192i·27-s + 0.371·29-s + 0.830·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.845173066\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.845173066\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.311 - 2.21i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 + 7.18iT - 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 - 6.23iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 - 4.85iT - 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 + 1.24iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 4.62iT - 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 0.488T + 61T^{2} \) |
| 67 | \( 1 - 3.61iT - 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 16.2iT - 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 - 8.23iT - 97T^{2} \) |
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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
−9.575199599485730602064472112723, −9.077758646062534145202306931835, −7.76933420356356800007574736206, −7.05684539044720951536632419713, −6.52705833955452526707345737382, −5.39747373645735920743117779802, −4.43686246709029978677315629674, −3.56202432856041743610982345722, −2.86349811258786289621682911177, −1.27109624379753640853342390219,
0.847910724087556684829637899944, 1.72951325100819977894398342628, 3.18745272688949554926810339142, 4.13077921822599670970705716554, 5.15601315600946275554836027872, 5.98974356769463674405524464335, 6.64712953142828354371986188107, 7.82625289499980702418707607753, 8.320086037480194555609848024004, 9.053968974486423362052174591297