| L(s) = 1 | + 2.82i·2-s − 8.00·4-s − 44.3i·5-s − 22.6i·8-s + 125.·10-s + 72.1i·11-s − 119.·13-s + 64.0·16-s + 166. i·17-s − 146.·19-s + 355. i·20-s − 204.·22-s − 586. i·23-s − 1.34e3·25-s − 337. i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.77i·5-s − 0.353i·8-s + 1.25·10-s + 0.596i·11-s − 0.705·13-s + 0.250·16-s + 0.577i·17-s − 0.404·19-s + 0.887i·20-s − 0.421·22-s − 1.10i·23-s − 2.15·25-s − 0.498i·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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2468804947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2468804947\) |
| \(L(3)\) |
|
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 | \( 1 + 44.3iT - 625T^{2} \) |
| 11 | \( 1 - 72.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 119.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 166. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 146.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 586. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.11e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.71e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.09e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.39e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 324.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.73e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.80e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.42e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.58e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.05e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.24e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.14e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.61e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.62e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.21e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.81e3T + 8.85e7T^{2} \) |
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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
−9.736451654327480799135095211120, −8.787828159650702804610772013411, −8.364354113088744584662711752354, −7.48188227196936292568154632991, −6.40129245246552781873888407373, −5.46788962006014647694205062957, −4.63434926807388323489765685132, −4.13119062655774435788510547511, −2.25325177881248342820739415507, −0.977678019291967596597671154417,
0.06173764590050240792605217879, 1.71574437937329526903066800115, 2.94071939860305833322443167712, 3.25841325844884859133815090180, 4.60612114154876683715917007880, 5.77018243728496983499025245202, 6.74471844759053026641740900864, 7.42452733739300504671676745694, 8.442060849016328013099929214706, 9.549829294043976605284075471279
