L(s) = 1 | + (1.79 + 1.79i)2-s + (−1.41 − 1.41i)3-s + 4.42i·4-s + (1.69 + 1.46i)5-s − 5.08i·6-s + (1.27 + 1.27i)7-s + (−4.35 + 4.35i)8-s + 1.01i·9-s + (0.412 + 5.65i)10-s + (6.27 − 6.27i)12-s + (−1.59 + 1.59i)13-s + 4.58i·14-s + (−0.326 − 4.47i)15-s − 6.74·16-s + (2.37 + 2.37i)17-s + (−1.82 + 1.82i)18-s + ⋯ |
L(s) = 1 | + (1.26 + 1.26i)2-s + (−0.818 − 0.818i)3-s + 2.21i·4-s + (0.756 + 0.653i)5-s − 2.07i·6-s + (0.483 + 0.483i)7-s + (−1.53 + 1.53i)8-s + 0.339i·9-s + (0.130 + 1.78i)10-s + (1.81 − 1.81i)12-s + (−0.442 + 0.442i)13-s + 1.22i·14-s + (−0.0842 − 1.15i)15-s − 1.68·16-s + (0.576 + 0.576i)17-s + (−0.430 + 0.430i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23555 + 2.14491i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23555 + 2.14491i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.69 - 1.46i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.79 - 1.79i)T + 2iT^{2} \) |
| 3 | \( 1 + (1.41 + 1.41i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1.27 - 1.27i)T + 7iT^{2} \) |
| 13 | \( 1 + (1.59 - 1.59i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.37 - 2.37i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 23 | \( 1 + (3.16 + 3.16i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 + (2.26 - 2.26i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.30iT - 41T^{2} \) |
| 43 | \( 1 + (-4.55 + 4.55i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.46 + 5.46i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.50 - 4.50i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 + 9.31iT - 61T^{2} \) |
| 67 | \( 1 + (2.46 - 2.46i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 + (-6.67 + 6.67i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.14T + 79T^{2} \) |
| 83 | \( 1 + (-7.39 + 7.39i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.85iT - 89T^{2} \) |
| 97 | \( 1 + (0.550 - 0.550i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.36014943714965729354033083968, −10.20141261542148192295948376652, −8.858843236365894226136785771982, −7.73953841257270583749934124678, −7.00567324042692751783810818033, −6.28845493578070223973843293622, −5.70239937879882407620680531478, −4.93583361530467638612044923531, −3.54593634912676991996107056341, −2.07336450671756479504494651237,
1.11155549253365341082348004484, 2.50455112390673274991996669797, 3.92594520879570385896784433299, 4.71843128783914305900174050336, 5.39338655130712196801112048185, 5.92959464919577924223424604735, 7.61097140970552453079888282660, 9.217511773048210580847628490355, 10.03850989022170532761258961072, 10.47308143723222031442725076517