L(s) = 1 | + i·2-s − 4-s + (1.41 + 1.73i)5-s − 1.03i·7-s − i·8-s + (−1.73 + 1.41i)10-s + 3.86·11-s − 1.03i·13-s + 1.03·14-s + 16-s − 7.46i·17-s + 19-s + (−1.41 − 1.73i)20-s + 3.86i·22-s − 1.46i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.632 + 0.774i)5-s − 0.391i·7-s − 0.353i·8-s + (−0.547 + 0.447i)10-s + 1.16·11-s − 0.287i·13-s + 0.276·14-s + 0.250·16-s − 1.81i·17-s + 0.229·19-s + (−0.316 − 0.387i)20-s + 0.823i·22-s − 0.305i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.992881465\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.992881465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.41 - 1.73i)T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 1.03iT - 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 + 1.03iT - 13T^{2} \) |
| 17 | \( 1 + 7.46iT - 17T^{2} \) |
| 23 | \( 1 + 1.46iT - 23T^{2} \) |
| 29 | \( 1 - 9.52T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 6.69iT - 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 1.79iT - 43T^{2} \) |
| 47 | \( 1 + 9.46iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 3.58iT - 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 13.3iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 4.39iT - 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−9.401020024019400224958548107202, −8.672605890701921395339003358991, −7.60230843437956142285627278560, −6.87970819332043650386342717377, −6.46506136709095214176852501726, −5.45452060855542491889013354948, −4.60733918705424592940343453681, −3.51890558221933551597862946451, −2.53368172209752490319801618265, −0.940057480850032100986087174772,
1.18942348892091815590722486608, 1.91997699699470899929889148461, 3.22341238503343690654358205928, 4.24976403894787311760930207382, 4.94771777850974583771724464556, 6.07907297265899307305344122702, 6.52034948272875569113125076239, 8.077921370227025993999669159633, 8.613393981711110045295796942671, 9.295259581837191283148781332024