L(s) = 1 | + (−1 + 1.41i)3-s + (2.23 − 1.41i)7-s + (−1.00 − 2.82i)9-s + 2.82i·11-s − 2.82i·13-s − 4·17-s − 6.32i·19-s + (−0.236 + 4.57i)21-s + 6.32i·23-s + (5.00 + 1.41i)27-s − 5.65i·29-s − 6.32i·31-s + (−4.00 − 2.82i)33-s − 8.94·37-s + (4.00 + 2.82i)39-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (0.845 − 0.534i)7-s + (−0.333 − 0.942i)9-s + 0.852i·11-s − 0.784i·13-s − 0.970·17-s − 1.45i·19-s + (−0.0515 + 0.998i)21-s + 1.31i·23-s + (0.962 + 0.272i)27-s − 1.05i·29-s − 1.13i·31-s + (−0.696 − 0.492i)33-s − 1.47·37-s + (0.640 + 0.452i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7621418722\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7621418722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 - 6.32iT - 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 6.32iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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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
−9.087690400898391079479064607155, −8.130754989241023510821439854685, −7.31266162343814268953820845971, −6.55570441124959709725569137623, −5.49810592099559209589744924158, −4.79951672695134963801350477065, −4.24726841395281649319856419949, −3.17486796212394515135986553542, −1.83415436008972393612715444535, −0.28731994301051079302390990365,
1.41748173979352115230614957211, 2.15627677858320513954708849280, 3.45110875732400806807329443058, 4.76552109722015745833079237049, 5.29537363513097536431220877157, 6.35367505677884087950366373125, 6.73184966724650541116701084692, 7.83208104239780186664027352006, 8.536677199761195456131364933167, 8.909367264878164647707441542092