L(s) = 1 | + (0.991 + 0.130i)3-s + (0.5 − 0.866i)4-s + (−0.923 − 1.60i)5-s + (0.965 + 0.258i)9-s + (0.866 + 0.5i)11-s + (0.608 − 0.793i)12-s + (−0.707 − 1.70i)15-s + (−0.499 − 0.866i)16-s − 1.84·20-s + (−1.22 + 0.707i)23-s + (−1.20 + 2.09i)25-s + (0.923 + 0.382i)27-s + (−0.662 − 0.382i)31-s + (0.793 + 0.608i)33-s + (0.707 − 0.707i)36-s + ⋯ |
L(s) = 1 | + (0.991 + 0.130i)3-s + (0.5 − 0.866i)4-s + (−0.923 − 1.60i)5-s + (0.965 + 0.258i)9-s + (0.866 + 0.5i)11-s + (0.608 − 0.793i)12-s + (−0.707 − 1.70i)15-s + (−0.499 − 0.866i)16-s − 1.84·20-s + (−1.22 + 0.707i)23-s + (−1.20 + 2.09i)25-s + (0.923 + 0.382i)27-s + (−0.662 − 0.382i)31-s + (0.793 + 0.608i)33-s + (0.707 − 0.707i)36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.539522652\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.539522652\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.991 - 0.130i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.866 - 0.5i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.84iT - T^{2} \) |
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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
−9.297003600271506705639028241551, −8.801186476892745747132814818767, −7.80797140291068612861743247855, −7.34517538131423497043541246646, −6.14253865311757705554762802187, −5.11371999822345674668576238140, −4.33501896753168290930205365978, −3.65157359019974881757209954375, −2.07011686585525507778927962588, −1.19839929814009191949122545134,
2.12293128000603020986841768624, 2.93594730491179742733655149823, 3.72635579400527284912329765786, 4.10121481379845189124671500227, 6.13302803642310233714656491753, 6.96048456722686662079087508398, 7.23361372560232326766158191802, 8.232745768090285323814015044723, 8.551265794368737837810177798151, 9.790810117474621776041648204534