L(s) = 1 | + (4.57 − 2.46i)3-s − 5i·5-s + 35.9i·7-s + (14.8 − 22.5i)9-s − 56.8·11-s + 48.7·13-s + (−12.3 − 22.8i)15-s + 18.2i·17-s − 20.4i·19-s + (88.5 + 164. i)21-s − 161.·23-s − 25·25-s + (12.4 − 139. i)27-s + 68.7i·29-s + 166. i·31-s + ⋯ |
L(s) = 1 | + (0.880 − 0.474i)3-s − 0.447i·5-s + 1.94i·7-s + (0.550 − 0.835i)9-s − 1.55·11-s + 1.03·13-s + (−0.212 − 0.393i)15-s + 0.260i·17-s − 0.246i·19-s + (0.920 + 1.70i)21-s − 1.46·23-s − 0.200·25-s + (0.0884 − 0.996i)27-s + 0.440i·29-s + 0.964i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.320540148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320540148\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.57 + 2.46i)T \) |
| 5 | \( 1 + 5iT \) |
good | 7 | \( 1 - 35.9iT - 343T^{2} \) |
| 11 | \( 1 + 56.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 20.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 68.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 166. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 135.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 32.0iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 259. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 396.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 477. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 268.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 501.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 327. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 945.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 376. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 506.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 944. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 643.T + 9.12e5T^{2} \) |
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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
−9.619344176820055400433897613129, −8.911463508991813454093952494774, −8.264820646609267897079655076135, −7.81851737108660376857913217323, −6.34296731567238309813759133087, −5.72784543187580635169252133249, −4.71811930227172975890172988992, −3.27045811459365971069996749654, −2.48836906370611224725758179071, −1.56429720055308410471991275545,
0.26877820870502509010531260975, 1.83860982707800935233942819015, 3.10607859420283341632086446680, 3.89599026442586113282735562258, 4.62450424185070434498214008025, 5.94923138859099772157936620733, 7.11002761031463967531650436559, 7.85580412184939062076842565422, 8.215475812467555887982317951081, 9.651108951941061479035461114710