L(s) = 1 | + (2.5 + 1.65i)3-s + 4·4-s − 9.94i·5-s + (3.5 + 8.29i)9-s + (10 + 6.63i)12-s + (16.5 − 24.8i)15-s + 16·16-s − 39.7i·20-s − 29.8i·23-s − 74·25-s + (−4.99 + 26.5i)27-s + 37·31-s + (14 + 33.1i)36-s + 25·37-s + (82.4 − 34.8i)45-s + ⋯ |
L(s) = 1 | + (0.833 + 0.552i)3-s + 4-s − 1.98i·5-s + (0.388 + 0.921i)9-s + (0.833 + 0.552i)12-s + (1.10 − 1.65i)15-s + 16-s − 1.98i·20-s − 1.29i·23-s − 2.95·25-s + (−0.185 + 0.982i)27-s + 1.19·31-s + (0.388 + 0.921i)36-s + 0.675·37-s + (1.83 − 0.773i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.65602 - 0.800820i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65602 - 0.800820i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.5 - 1.65i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 4T^{2} \) |
| 5 | \( 1 + 9.94iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 + 29.8iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 37T + 961T^{2} \) |
| 37 | \( 1 - 25T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 79.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 79.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 49.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 35T + 4.48e3T^{2} \) |
| 71 | \( 1 - 49.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 149. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 95T + 9.40e3T^{2} \) |
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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
−11.10753404738016403761117123716, −10.02480569420035555321838446259, −9.221294436478996552058921796469, −8.326368031668718769837317867305, −7.75687021152315944090968060667, −6.20945163361153676901049389246, −4.99063087430731484430696632607, −4.16676985653622901857812369999, −2.61752170669671940248949119837, −1.27822809468157391260321744731,
1.91432570704106936968798957549, 2.88002660931729143634468657522, 3.60283019630482592464062043948, 5.98238760160443418989304343754, 6.75144386681675944432626640445, 7.35938965037992557991093614051, 8.128504384806102727876891841569, 9.715510668862999087730019259875, 10.36640137621970247253454740722, 11.40705187730613214962147884641