| L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 25.7i·5-s + 22.6i·8-s + 72.9·10-s + 2.36i·11-s − 268.·13-s + 64.0·16-s − 258. i·17-s + 527.·19-s − 206. i·20-s + 6.69·22-s − 128. i·23-s − 39.7·25-s + 759. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.03i·5-s + 0.353i·8-s + 0.729·10-s + 0.0195i·11-s − 1.58·13-s + 0.250·16-s − 0.894i·17-s + 1.46·19-s − 0.515i·20-s + 0.0138·22-s − 0.242i·23-s − 0.0636·25-s + 1.12i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.468302783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468302783\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 25.7iT - 625T^{2} \) |
| 11 | \( 1 - 2.36iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 268.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 258. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 527.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 128. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.45e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 792.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.00e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.01e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.49e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 655. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.82e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.91e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.24e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.69e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 939.T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.39e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.99e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 4.12e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.97e3T + 8.85e7T^{2} \) |
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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
−9.702372871509513982421683274104, −9.244786940645995889233831310299, −7.65065890236857677442396987809, −7.40355312232954258280602200725, −6.15600731871106509744549408553, −5.11086956115733166500208382043, −4.16042983190652287434912828946, −2.83991295281292777573236469626, −2.50157634157362136974431595541, −0.850002173691903150621085641133,
0.42457372723586199945574482669, 1.61913242631503997665678965728, 3.16765652794230976680525668307, 4.43447009341900020971459960089, 5.14514393642418144136596130958, 5.83394907375191456863215484036, 7.15676299221572767869082249171, 7.64194801417348623520231374980, 8.676831503105978731280747916981, 9.315892891712157109602737202205
