L(s) = 1 | + (0.490 + 1.32i)2-s + (1.99 − 1.99i)3-s + (−1.51 + 1.30i)4-s + (−2.16 + 0.569i)5-s + (3.61 + 1.66i)6-s + 1.09·7-s + (−2.46 − 1.37i)8-s − 4.93i·9-s + (−1.81 − 2.58i)10-s + (−2.33 + 2.33i)11-s + (−0.437 + 5.61i)12-s + (−1.80 + 1.80i)13-s + (0.534 + 1.44i)14-s + (−3.17 + 5.44i)15-s + (0.619 − 3.95i)16-s − 4.93i·17-s + ⋯ |
L(s) = 1 | + (0.346 + 0.938i)2-s + (1.14 − 1.14i)3-s + (−0.759 + 0.650i)4-s + (−0.966 + 0.254i)5-s + (1.47 + 0.680i)6-s + 0.412·7-s + (−0.873 − 0.487i)8-s − 1.64i·9-s + (−0.574 − 0.818i)10-s + (−0.703 + 0.703i)11-s + (−0.126 + 1.62i)12-s + (−0.501 + 0.501i)13-s + (0.142 + 0.386i)14-s + (−0.818 + 1.40i)15-s + (0.154 − 0.987i)16-s − 1.19i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19607 + 0.302138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19607 + 0.302138i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.490 - 1.32i)T \) |
| 5 | \( 1 + (2.16 - 0.569i)T \) |
good | 3 | \( 1 + (-1.99 + 1.99i)T - 3iT^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 + (2.33 - 2.33i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.80 - 1.80i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.93iT - 17T^{2} \) |
| 19 | \( 1 + (-2.03 - 2.03i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 + (0.707 + 0.707i)T + 29iT^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + (4.35 + 4.35i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (-2.22 - 2.22i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.09iT - 47T^{2} \) |
| 53 | \( 1 + (0.215 + 0.215i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.16 - 1.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.46 + 3.46i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.04 + 5.04i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.40iT - 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 - 2.61T + 79T^{2} \) |
| 83 | \( 1 + (5.67 - 5.67i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.87iT - 89T^{2} \) |
| 97 | \( 1 - 3.77iT - 97T^{2} \) |
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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
−14.40597191930621962980458603190, −13.72639447222628438449124209202, −12.57071896219854232457690576546, −11.79697692591194789312303722776, −9.538337964070882368457714740061, −8.159677949191236158200602711790, −7.64242927566893300050131230242, −6.76949737527615949497877825070, −4.65206840014813268123618430060, −2.90833175246742863955935257584,
2.92195415866566169542857055225, 4.03081793542534553811673865677, 5.14916230250929357164987636593, 8.087125105082376760266939909266, 8.716468017906150011604131822164, 10.08210207349366531212762223384, 10.84758415589388918929266661413, 12.07724161997892262102972669061, 13.34308007879093733475235366202, 14.32884406977739065431452367554