L(s) = 1 | + 2i·2-s − 4·4-s + (−10 + 5i)5-s − 7i·7-s − 8i·8-s + (−10 − 20i)10-s + 37·11-s + 51i·13-s + 14·14-s + 16·16-s + 41i·17-s + 108·19-s + (40 − 20i)20-s + 74i·22-s + 70i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.377i·7-s − 0.353i·8-s + (−0.316 − 0.632i)10-s + 1.01·11-s + 1.08i·13-s + 0.267·14-s + 0.250·16-s + 0.584i·17-s + 1.30·19-s + (0.447 − 0.223i)20-s + 0.717i·22-s + 0.634i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6372723578\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6372723578\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (10 - 5i)T \) |
| 7 | \( 1 + 7iT \) |
good | 11 | \( 1 - 37T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 41iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 108T + 6.85e3T^{2} \) |
| 23 | \( 1 - 70iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 249T + 2.43e4T^{2} \) |
| 31 | \( 1 + 134T + 2.97e4T^{2} \) |
| 37 | \( 1 - 334iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 206T + 6.89e4T^{2} \) |
| 43 | \( 1 + 376iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 287iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 6iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 940T + 2.26e5T^{2} \) |
| 67 | \( 1 + 106iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 456T + 3.57e5T^{2} \) |
| 73 | \( 1 - 650iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 428iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 220T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3iT - 9.12e5T^{2} \) |
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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
−10.72603460311465949298421503942, −9.597723325646776451877683842247, −8.915062921087237537471510431304, −7.84106422367758109736292742404, −7.13224424003587041105373076634, −6.49550488558850447515671190071, −5.26258834379318647545066213802, −4.04885445611249140480425287445, −3.49510918714128743895266790079, −1.48231077625781851414051174005,
0.20234933933853706015688733059, 1.38473405194428741521567792849, 2.99028475327962788497942208885, 3.81825252562997186121760940613, 4.90875091422231700969318369327, 5.81325642214351136113544996696, 7.29214796333505267818962512698, 7.989100203037682398572975020342, 9.103949994504107886569059104777, 9.489407470476623909613195808974