L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.46 + 0.931i)3-s − 1.00i·4-s + (−2.16 + 0.569i)5-s + (−1.69 + 0.373i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (1.26 + 2.72i)9-s + (1.12 − 1.93i)10-s + 6.30i·11-s + (0.931 − 1.46i)12-s + (−0.977 + 0.977i)13-s − 1.00·14-s + (−3.68 − 1.18i)15-s − 1.00·16-s + (4.86 − 4.86i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.843 + 0.537i)3-s − 0.500i·4-s + (−0.967 + 0.254i)5-s + (−0.690 + 0.152i)6-s + (0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (0.421 + 0.906i)9-s + (0.356 − 0.610i)10-s + 1.90i·11-s + (0.268 − 0.421i)12-s + (−0.271 + 0.271i)13-s − 0.267·14-s + (−0.952 − 0.305i)15-s − 0.250·16-s + (1.18 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.650801 + 0.831915i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.650801 + 0.831915i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-1.46 - 0.931i)T \) |
| 5 | \( 1 + (2.16 - 0.569i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 6.30iT - 11T^{2} \) |
| 13 | \( 1 + (0.977 - 0.977i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.86 + 4.86i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.285iT - 19T^{2} \) |
| 23 | \( 1 + (4.26 + 4.26i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 + (-0.317 - 0.317i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.55iT - 41T^{2} \) |
| 43 | \( 1 + (-2.07 + 2.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.69 + 6.69i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.12 + 3.12i)T + 53iT^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 1.09T + 61T^{2} \) |
| 67 | \( 1 + (5.63 + 5.63i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.42iT - 71T^{2} \) |
| 73 | \( 1 + (3.69 - 3.69i)T - 73iT^{2} \) |
| 79 | \( 1 - 4.38iT - 79T^{2} \) |
| 83 | \( 1 + (-1.52 - 1.52i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 + (-1.50 - 1.50i)T + 97iT^{2} \) |
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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
−12.49151645271447371369868681506, −11.69436023916355813629111712188, −10.27406274055473656237197195184, −9.719713847118390238750510773161, −8.592267560281226522584069571405, −7.63203400309447093943334362870, −7.07093047251605672197020029197, −5.06848253356493130549769466373, −4.10998428543784170438159171214, −2.38455006758663519176797024465,
1.08155541452044092704880504932, 3.11413101731686034323565811448, 3.94392799256593053201899564954, 5.98278647239569950377539705068, 7.62711130896001114257964355923, 8.086257080478844863485375772677, 8.860014417537472522185256516715, 10.12524961422174880031592341052, 11.24176805756638669200383665996, 12.01820906540380745010835197657