| L(s) = 1 | + 5.33i·2-s − 12.4·4-s + 39.7i·5-s + 65.7·7-s + 18.7i·8-s − 212.·10-s − 38.0i·11-s − 240.·13-s + 350. i·14-s − 299.·16-s − 151. i·17-s + 313.·19-s − 496. i·20-s + 202.·22-s + 327. i·23-s + ⋯ |
| L(s) = 1 | + 1.33i·2-s − 0.780·4-s + 1.59i·5-s + 1.34·7-s + 0.293i·8-s − 2.12·10-s − 0.314i·11-s − 1.42·13-s + 1.78i·14-s − 1.17·16-s − 0.525i·17-s + 0.867·19-s − 1.24i·20-s + 0.419·22-s + 0.618i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(-1.70677i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-1.70677i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 5.33iT - 16T^{2} \) |
| 5 | \( 1 - 39.7iT - 625T^{2} \) |
| 7 | \( 1 - 65.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 38.0iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 240.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 151. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 313.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 327. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 484. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 82.5T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.78e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.68e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.88e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.75e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 540. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 882. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.30e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.95e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.73e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.97e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.06e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.27e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.37e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.04e3T + 8.85e7T^{2} \) |
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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
−14.51808691039735934599085113789, −13.81378129045460410843865889856, −11.69566469333759163697880984789, −11.01350962532666030562469666255, −9.552469549726037079928970086200, −7.75678155181317026038775698675, −7.43292661217934052072340651471, −6.09652593640140970843979347546, −4.81861013138496693311279922383, −2.58256876055116915801654934688,
0.896130973212881351912478947986, 2.12576082865448260325058197383, 4.31968377378771400497967575968, 5.15302440268714971060667593008, 7.60532288508656407865824291196, 8.843913835022245574351503245541, 9.828513964629986537960759472598, 11.07239553953431048938643797016, 12.19135810565096422239732666555, 12.50624831464449368923981311574
