L(s) = 1 | + 3·9-s + 6i·13-s + 2i·17-s − 10·29-s + 2i·37-s + 10·41-s + 7·49-s + 14i·53-s + 10·61-s + 6i·73-s + 9·81-s − 10·89-s + 18i·97-s + 2·101-s + 6·109-s + ⋯ |
L(s) = 1 | + 9-s + 1.66i·13-s + 0.485i·17-s − 1.85·29-s + 0.328i·37-s + 1.56·41-s + 49-s + 1.92i·53-s + 1.28·61-s + 0.702i·73-s + 81-s − 1.05·89-s + 1.82i·97-s + 0.199·101-s + 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.658334805\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658334805\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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
−9.379985788756507890248421986987, −9.016694792407669942144884550905, −7.81098404526705947035644607784, −7.16211540055356908696640317727, −6.42305597257860151217384376818, −5.47609993860249359551185515327, −4.29583604653231160094339269388, −3.90274935645140785862603538659, −2.34529288830870757534092297515, −1.37774851511004780680395899227,
0.68967941519025122210458517661, 2.09752032955479824021626459538, 3.28514445811830652857474085937, 4.16525659046008017832538891650, 5.24094691624264550546619660031, 5.87475744794244571478260991634, 7.07700643386195257397833728416, 7.57493599446690188721245822483, 8.400361364554562061744997916979, 9.421389577647872954165604973444