L(s) = 1 | + (1.20 + 1.20i)2-s + 0.924i·4-s + (1.20 + 1.88i)5-s + (2.18 − 2.18i)7-s + (1.30 − 1.30i)8-s + (−0.812 + 3.73i)10-s − 6.39i·11-s + (3.18 + 3.18i)13-s + 5.28·14-s + 4.99·16-s + (−4.92 − 4.92i)17-s + i·19-s + (−1.73 + 1.11i)20-s + (7.73 − 7.73i)22-s + (−1.62 + 1.62i)23-s + ⋯ |
L(s) = 1 | + (0.855 + 0.855i)2-s + 0.462i·4-s + (0.540 + 0.841i)5-s + (0.825 − 0.825i)7-s + (0.459 − 0.459i)8-s + (−0.256 + 1.18i)10-s − 1.92i·11-s + (0.883 + 0.883i)13-s + 1.41·14-s + 1.24·16-s + (−1.19 − 1.19i)17-s + 0.229i·19-s + (−0.388 + 0.250i)20-s + (1.64 − 1.64i)22-s + (−0.338 + 0.338i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.85234 + 0.934659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.85234 + 0.934659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.20 - 1.88i)T \) |
| 19 | \( 1 - iT \) |
good | 2 | \( 1 + (-1.20 - 1.20i)T + 2iT^{2} \) |
| 7 | \( 1 + (-2.18 + 2.18i)T - 7iT^{2} \) |
| 11 | \( 1 + 6.39iT - 11T^{2} \) |
| 13 | \( 1 + (-3.18 - 3.18i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.92 + 4.92i)T + 17iT^{2} \) |
| 23 | \( 1 + (1.62 - 1.62i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 + (6.56 - 6.56i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.32iT - 41T^{2} \) |
| 43 | \( 1 + (-4.02 - 4.02i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.10 + 2.10i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.08 - 7.08i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.04T + 59T^{2} \) |
| 61 | \( 1 + 2.29T + 61T^{2} \) |
| 67 | \( 1 + (3.38 - 3.38i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (-5.68 - 5.68i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.156iT - 79T^{2} \) |
| 83 | \( 1 + (-8.12 + 8.12i)T - 83iT^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + (-0.439 + 0.439i)T - 97iT^{2} \) |
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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
−10.52098462622647723624578600531, −9.351211110930155458419352540592, −8.352715556417986367290285355033, −7.41541393132116901666031715707, −6.54327097945996898311022595874, −6.09276265664119798104130591156, −5.04185577555209907823464398900, −4.09199304396202806450300150437, −3.11004102059046477477080273261, −1.35886906501131070552840472171,
1.81471694074854427549700048966, 2.16159398193072171034659577413, 3.83440468046013658596891285774, 4.71845617138921207802711717562, 5.25974632046301523454785548616, 6.30093950836340200529005110252, 7.73989850335794677437675319779, 8.505867066083111839702900982630, 9.254316158374347131083899679574, 10.45468669835365243717772697222