| L(s) = 1 | + (−885. − 511. i)5-s + (765. − 2.27e3i)7-s + (−89.3 − 154. i)11-s + 1.94e4i·13-s + (−6.42e4 + 3.70e4i)17-s + (4.26e3 + 2.46e3i)19-s + (−9.62e4 + 1.66e5i)23-s + (3.27e5 + 5.66e5i)25-s − 1.17e6·29-s + (−9.50e5 + 5.48e5i)31-s + (−1.84e6 + 1.62e6i)35-s + (1.46e6 − 2.54e6i)37-s − 7.19e4i·41-s + 6.34e5·43-s + (−1.90e5 − 1.09e5i)47-s + ⋯ |
| L(s) = 1 | + (−1.41 − 0.817i)5-s + (0.318 − 0.947i)7-s + (−0.00610 − 0.0105i)11-s + 0.681i·13-s + (−0.769 + 0.444i)17-s + (0.0327 + 0.0189i)19-s + (−0.343 + 0.595i)23-s + (0.837 + 1.45i)25-s − 1.66·29-s + (−1.02 + 0.593i)31-s + (−1.22 + 1.08i)35-s + (0.783 − 1.35i)37-s − 0.0254i·41-s + 0.185·43-s + (−0.0389 − 0.0224i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0724i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.997 - 0.0724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9338268639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9338268639\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-765. + 2.27e3i)T \) |
| good | 5 | \( 1 + (885. + 511. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (89.3 + 154. i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 1.94e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (6.42e4 - 3.70e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-4.26e3 - 2.46e3i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (9.62e4 - 1.66e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.17e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (9.50e5 - 5.48e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.46e6 + 2.54e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 7.19e4iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.34e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (1.90e5 + 1.09e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-6.01e5 - 1.04e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-8.57e6 + 4.95e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.33e7 - 7.70e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-9.22e5 - 1.59e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.59e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.11e7 + 6.42e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.23e7 - 3.86e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 2.52e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (5.07e7 + 2.92e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.08e8iT - 7.83e15T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.01147345398216580045890813077, −9.521258342548337298045825485713, −8.581567173102749935354641609170, −7.68126853730255374835106791206, −6.99092584463389663857709904120, −5.36682375841447267619864132507, −4.15184428511424932179217769199, −3.80234278630374179146072799589, −1.79103141497659325754498229416, −0.58062588308573346339374647507,
0.35612688395861300901702705508, 2.21447798665596877342403495305, 3.24009698429894731295256212327, 4.30675555417440567247810910183, 5.55432566211602780492888430017, 6.78153941260608292183331917341, 7.74295455009353760180871002786, 8.456587537720288265088491146351, 9.619208145695646875366006105971, 10.97794958185185058331805260238
