L(s) = 1 | + 3·3-s + 7.52i·5-s + (−4.86 − 17.8i)7-s + 9·9-s + 29.4i·11-s − 5.35i·13-s + 22.5i·15-s + 66.3i·17-s + 22.3·19-s + (−14.5 − 53.6i)21-s − 33.9i·23-s + 68.4·25-s + 27·27-s + 133.·29-s − 323.·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.672i·5-s + (−0.262 − 0.964i)7-s + 0.333·9-s + 0.806i·11-s − 0.114i·13-s + 0.388i·15-s + 0.946i·17-s + 0.270·19-s + (−0.151 − 0.557i)21-s − 0.308i·23-s + 0.547·25-s + 0.192·27-s + 0.853·29-s − 1.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.808453704\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.808453704\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + (4.86 + 17.8i)T \) |
good | 5 | \( 1 - 7.52iT - 125T^{2} \) |
| 11 | \( 1 - 29.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 5.35iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 66.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 22.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 33.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 323.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 140. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 19.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 376.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 441.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 130. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 627. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 808. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 417. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 214. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 639.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 686. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 103. iT - 9.12e5T^{2} \) |
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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
−9.585963177524805868688974427165, −8.629791110185959452228377838460, −7.76930851973165972083844076930, −7.03121659354392157491147679617, −6.51879641561279555913152225395, −5.20547468943233289391147741665, −4.12057373921762624991625816604, −3.44821664774120722684450935559, −2.38981859193585531062676115500, −1.22877301237552752752414669866,
0.38421126551438276588426226576, 1.70886219594804867808869809872, 2.82817607839762233793645514779, 3.61406234362829064397381788106, 4.91894690402429978130214196702, 5.50760218843856002363540393489, 6.55058343570103769090172677985, 7.49465846248841026963027598881, 8.439115855518333061613837183062, 8.996355081521786207358894635897