| L(s) = 1 | + 3.46i·5-s − 24.2·7-s − 48i·11-s + 41.5i·13-s − 54·17-s + 4i·19-s + 173.·23-s + 113·25-s − 162. i·29-s − 58.8·31-s − 84i·35-s + 325. i·37-s + 294·41-s + 188i·43-s + 505.·47-s + ⋯ |
| L(s) = 1 | + 0.309i·5-s − 1.30·7-s − 1.31i·11-s + 0.886i·13-s − 0.770·17-s + 0.0482i·19-s + 1.57·23-s + 0.904·25-s − 1.04i·29-s − 0.341·31-s − 0.405i·35-s + 1.44i·37-s + 1.11·41-s + 0.666i·43-s + 1.56·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.481489787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481489787\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46iT - 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 + 48iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 294T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 744. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 252iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 90.0iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 720iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−10.47953250264759383745345065547, −9.219194607910749997618333074596, −8.953827163348471732653372552728, −7.57301328553030623023556882839, −6.53193620338873006370821076968, −6.11132725978063392415795217119, −4.66864762178419994200437151574, −3.44629429571768824482172349234, −2.64614978469785533272485355759, −0.76867519395063296878116060024,
0.67351929185989974637924907125, 2.38083665624231994010067269951, 3.47413123486996255573483745036, 4.68495035856480633254807283262, 5.64635638138415611073676200182, 6.87561346093139081682795401885, 7.31945038217256939267313705900, 8.820170262956957910159923995391, 9.296994944918688077597350850300, 10.29065678981193123242710098143
