L(s) = 1 | + 2.67i·2-s − 5.13·4-s + 3.20·5-s − 8.37i·8-s + 8.55i·10-s − 4.06i·11-s − 2.99i·13-s + 12.0·16-s + 2.54·17-s + 0.165i·19-s − 16.4·20-s + 10.8·22-s − 4.04i·23-s + 5.26·25-s + 8.00·26-s + ⋯ |
L(s) = 1 | + 1.88i·2-s − 2.56·4-s + 1.43·5-s − 2.95i·8-s + 2.70i·10-s − 1.22i·11-s − 0.831i·13-s + 3.02·16-s + 0.616·17-s + 0.0379i·19-s − 3.67·20-s + 2.31·22-s − 0.843i·23-s + 1.05·25-s + 1.57·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.763135246\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.763135246\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.67iT - 2T^{2} \) |
| 5 | \( 1 - 3.20T + 5T^{2} \) |
| 11 | \( 1 + 4.06iT - 11T^{2} \) |
| 13 | \( 1 + 2.99iT - 13T^{2} \) |
| 17 | \( 1 - 2.54T + 17T^{2} \) |
| 19 | \( 1 - 0.165iT - 19T^{2} \) |
| 23 | \( 1 + 4.04iT - 23T^{2} \) |
| 29 | \( 1 - 2.12iT - 29T^{2} \) |
| 31 | \( 1 + 7.70iT - 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 0.629T + 41T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 - 2.48iT - 53T^{2} \) |
| 59 | \( 1 - 8.84T + 59T^{2} \) |
| 61 | \( 1 - 10.8iT - 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 + 9.33iT - 71T^{2} \) |
| 73 | \( 1 - 6.10iT - 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−9.616115835023916139679908551689, −8.737335623231802020425733255300, −8.146268921553860353416428668491, −7.30766264815843300002758601965, −6.20289103669361852511290104530, −5.90889862889434803073879106192, −5.30019487451808637572094064702, −4.18373421213912666596018209716, −2.82757764577683708240378394099, −0.834972320083100621018638902163,
1.36457589187075634776528072207, 2.02554443854544269464578139935, 2.91650611465486858632038798846, 4.12158319528299235248350029533, 4.93653002888452552319981718442, 5.76687516494144161371291088986, 6.97126805046528392790761853142, 8.243385919695035663776264127156, 9.271383013904942983244364118819, 9.724583326681731535250837492874