L(s) = 1 | + i·2-s + 3-s − 4-s + (2 − i)5-s + i·6-s + 2·7-s − i·8-s − 2·9-s + (1 + 2i)10-s − 12-s + i·13-s + 2i·14-s + (2 − i)15-s + 16-s + (1 + 4i)17-s − 2i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + (0.894 − 0.447i)5-s + 0.408i·6-s + 0.755·7-s − 0.353i·8-s − 0.666·9-s + (0.316 + 0.632i)10-s − 0.288·12-s + 0.277i·13-s + 0.534i·14-s + (0.516 − 0.258i)15-s + 0.250·16-s + (0.242 + 0.970i)17-s − 0.471i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35518 + 0.501313i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35518 + 0.501313i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 17 | \( 1 + (-1 - 4i)T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 - 5iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−13.20003916449535018258867736667, −12.09850529905754912727863072647, −10.71369463785094807460861887954, −9.595427966477503798189291150409, −8.553894901425455625758113621537, −8.051648952256609540644552125398, −6.40224127653083741685001868836, −5.48830190311268484103928206059, −4.14800988415631466917326004810, −2.09774505601165615279768778452,
2.01602626977355442406190385855, 3.16218883380341411967148272410, 4.87878126295597667599082294631, 6.12219164428167399966994400260, 7.73819026466125478177770246183, 8.777267495738734926545470281019, 9.673699245760630854857163727186, 10.73606964720735632063528440668, 11.49070460681623677730087814455, 12.73069282427223356333378913942