| L(s) = 1 | + 2-s + 4-s + (−0.613 + 2.15i)5-s + 8-s + (−0.613 + 2.15i)10-s − 2.77i·11-s + 4.99·13-s + 16-s − 4.36i·17-s + 1.15i·19-s + (−0.613 + 2.15i)20-s − 2.77i·22-s + 2.40·23-s + (−4.24 − 2.63i)25-s + 4.99·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.274 + 0.961i)5-s + 0.353·8-s + (−0.193 + 0.680i)10-s − 0.837i·11-s + 1.38·13-s + 0.250·16-s − 1.05i·17-s + 0.264i·19-s + (−0.137 + 0.480i)20-s − 0.592i·22-s + 0.501·23-s + (−0.849 − 0.527i)25-s + 0.978·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.167842167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.167842167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.613 - 2.15i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.77iT - 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 + 4.36iT - 17T^{2} \) |
| 19 | \( 1 - 1.15iT - 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 6.90iT - 29T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 + 0.263iT - 37T^{2} \) |
| 41 | \( 1 - 6.42T + 41T^{2} \) |
| 43 | \( 1 + 2.17iT - 43T^{2} \) |
| 47 | \( 1 + 6.93iT - 47T^{2} \) |
| 53 | \( 1 - 5.34T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 8.51iT - 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 4.98iT - 71T^{2} \) |
| 73 | \( 1 + 9.07T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 9.01iT - 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
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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
−8.293588672722367659709026064351, −7.41191995283627549806285689167, −6.84048722936585389861426492427, −6.04926896042203827364175712226, −5.59245838308351372667625429113, −4.47499015371649403267065849630, −3.65257608322213264477002844562, −3.12471060847262231040671329947, −2.23626439611471363581309741031, −0.826063299493982123892679842900,
1.06760152183946399198822269045, 1.88813151770201237996563797862, 3.12820626905557245109616615055, 4.07796602738643321862936008057, 4.43206838477938788607186813535, 5.44538632573412743873235038953, 5.95921540428661893361488256771, 6.85922123520044099670691760962, 7.59862578309145174342368385755, 8.411793219858608580750005368247
