| L(s) = 1 | + (−1.75 + 3.59i)2-s + (−9.84 − 12.6i)4-s + 46.6·5-s + 60.5i·7-s + (62.6 − 13.2i)8-s + (−81.9 + 167. i)10-s + 73.6i·11-s − 31.1·13-s + (−217. − 106. i)14-s + (−62.2 + 248. i)16-s + 53.8·17-s − 54.9i·19-s + (−459. − 589. i)20-s + (−264. − 129. i)22-s + 281. i·23-s + ⋯ |
| L(s) = 1 | + (−0.438 + 0.898i)2-s + (−0.615 − 0.788i)4-s + 1.86·5-s + 1.23i·7-s + (0.978 − 0.206i)8-s + (−0.819 + 1.67i)10-s + 0.608i·11-s − 0.184·13-s + (−1.11 − 0.542i)14-s + (−0.243 + 0.969i)16-s + 0.186·17-s − 0.152i·19-s + (−1.14 − 1.47i)20-s + (−0.546 − 0.266i)22-s + 0.532i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.007471817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007471817\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.75 - 3.59i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 46.6T + 625T^{2} \) |
| 7 | \( 1 - 60.5iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 73.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 31.1T + 2.85e4T^{2} \) |
| 17 | \( 1 - 53.8T + 8.35e4T^{2} \) |
| 19 | \( 1 + 54.9iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 281. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 447.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 277. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.01e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.89e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 769. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.74e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.64e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 303. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 956.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.94e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.97e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.33e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.80e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.15e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.13e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.96e3T + 8.85e7T^{2} \) |
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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
−11.00989911531193486999927941714, −9.869243519664426269632954790279, −9.441059827792387776008855913960, −8.710975813829412226988793288475, −7.39733264953367177018717810519, −6.22098763272688644818112399987, −5.69853520096722251537511020577, −4.82191319360758231281827956072, −2.50375107498926837624772315299, −1.47147510413788594077423114865,
0.72393605848582869188956677459, 1.80151765294612462596472454507, 2.99609349841148681587356673717, 4.37427564050582932624456299720, 5.63008530192975342798278916201, 6.79531384239577347337532467724, 7.989866729164433461260193973613, 9.174363856202915637380133080926, 9.837501735718840780242456674227, 10.53040762082929532296942454185
