L(s) = 1 | + 2i·2-s − 4·4-s − 30.5i·7-s − 8i·8-s + 13.5·11-s − 28.0i·13-s + 61.0·14-s + 16·16-s + 55.5i·17-s − 27.4·19-s + 27.0i·22-s + 139. i·23-s + 56.1·26-s + 122. i·28-s + 178.·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.64i·7-s − 0.353i·8-s + 0.371·11-s − 0.598i·13-s + 1.16·14-s + 0.250·16-s + 0.792i·17-s − 0.331·19-s + 0.262i·22-s + 1.26i·23-s + 0.423·26-s + 0.824i·28-s + 1.14·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.597111317\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597111317\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 30.5iT - 343T^{2} \) |
| 11 | \( 1 - 13.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 55.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 27.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 139. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 159. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.68iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 301. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 122. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 864.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 47.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 402. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 927.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 812.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 124. iT - 9.12e5T^{2} \) |
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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
−8.974607036114580543571536097291, −8.021796999962232571565362775687, −7.53234550639368314353757747273, −6.65178559811378312509060454824, −5.98588461216421299500132934708, −4.78243167355855273930496848019, −4.08526124839224354621411854239, −3.20921362662727138148822446006, −1.42051145464607187768061029117, −0.42320513906007577364256908561,
1.10271328391199555220210371974, 2.42793588831242688516741478156, 2.85205671214854952425656185500, 4.33098216813744037434418802622, 4.98085640457811953965922886171, 6.08243671316536853724816550295, 6.71148630550943429114019056254, 8.176149585210469790539796208312, 8.660517988272816931335399810078, 9.414748682147728531454511180109