L(s) = 1 | − 12.7·2-s + 98·4-s + 524i·7-s − 432.·8-s − 865. i·11-s − 344i·13-s − 6.66e3i·14-s − 763.·16-s + 7.14e3·17-s + 2.32e3·19-s + 1.10e4i·22-s + 5.75e3·23-s + 4.37e3i·26-s + 5.13e4i·28-s − 2.31e4i·29-s + ⋯ |
L(s) = 1 | − 1.59·2-s + 1.53·4-s + 1.52i·7-s − 0.845·8-s − 0.650i·11-s − 0.156i·13-s − 2.43i·14-s − 0.186·16-s + 1.45·17-s + 0.338·19-s + 1.03i·22-s + 0.472·23-s + 0.249i·26-s + 2.33i·28-s − 0.949i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8869150181\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8869150181\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 12.7T + 64T^{2} \) |
| 7 | \( 1 - 524iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 865. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 344iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 7.14e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 2.32e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 5.75e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.31e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.05e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.40e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.08e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 9.09e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.28e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.96e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.98e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.51e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.16e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 5.39e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.08e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 5.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.32e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 2.23e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.71e4iT - 8.32e11T^{2} \) |
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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
−11.06060561419739447590228216397, −9.903584928060381960035272143740, −9.226866684899245056664498753011, −8.376843521823057055672107199773, −7.65005356753032898997342082433, −6.26836445758241770436698999008, −5.29666100369288907621269424907, −3.15423507319884356064438414774, −1.96139461768154734533596222237, −0.64304368272510445362205949721,
0.74073087063230340841642279208, 1.58203512297577123235059938342, 3.38119686867549761736127228659, 4.84754586912347575020618429568, 6.66220865799617902940044168686, 7.42450178285223705029924955414, 8.092175974600731850434021403636, 9.392581790985938101434936330302, 10.06909456465117416414799163468, 10.72804504166127451264512909121