L(s) = 1 | − 2.28i·3-s − 0.540·5-s + (1.41 − 2.23i)7-s − 2.23·9-s − 1.23·11-s + 2.28·13-s + 1.23i·15-s + 1.74i·17-s − 5.11i·19-s + (−5.11 − 3.23i)21-s − 3.23i·23-s − 4.70·25-s − 1.74i·27-s − 2i·29-s + 3.90·31-s + ⋯ |
L(s) = 1 | − 1.32i·3-s − 0.241·5-s + (0.534 − 0.845i)7-s − 0.745·9-s − 0.372·11-s + 0.634·13-s + 0.319i·15-s + 0.423i·17-s − 1.17i·19-s + (−1.11 − 0.706i)21-s − 0.674i·23-s − 0.941·25-s − 0.336i·27-s − 0.371i·29-s + 0.702·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.398719957\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.398719957\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
good | 3 | \( 1 + 2.28iT - 3T^{2} \) |
| 5 | \( 1 + 0.540T + 5T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 - 1.74iT - 17T^{2} \) |
| 19 | \( 1 + 5.11iT - 19T^{2} \) |
| 23 | \( 1 + 3.23iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 3.90T + 31T^{2} \) |
| 37 | \( 1 + 6.94iT - 37T^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 + 1.62iT - 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 - 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 3.52iT - 79T^{2} \) |
| 83 | \( 1 - 9.02iT - 83T^{2} \) |
| 89 | \( 1 + 15.4iT - 89T^{2} \) |
| 97 | \( 1 + 1.74iT - 97T^{2} \) |
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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
−8.484800825030232914788338044570, −8.093723479592920977545173093123, −7.30708252475458573958340097651, −6.71599168439055060373626327951, −5.92455038270486617788597265545, −4.75985048805618659140332779152, −3.90464961082538248215340959227, −2.61819394226843485073210836770, −1.57295057595404064724560047944, −0.52895958142940019983078819058,
1.70542610278770498000751589968, 3.06550494943910440800723745661, 3.85028593787099717454773798894, 4.73823508730671131058442269365, 5.45030714280620530604348071952, 6.15704191091638781890169202354, 7.50233097135175529763578820415, 8.232599396555703072698981341878, 8.973393894484589606922415298935, 9.606658128841806032412615179550