| L(s) = 1 | − i·3-s + 3.46·7-s + 2·9-s − 3i·11-s − 3.46i·13-s − 3·17-s − i·19-s − 3.46i·21-s − 5i·27-s + 10.3i·29-s + 6.92·31-s − 3·33-s − 10.3i·37-s − 3.46·39-s − 9·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.30·7-s + 0.666·9-s − 0.904i·11-s − 0.960i·13-s − 0.727·17-s − 0.229i·19-s − 0.755i·21-s − 0.962i·27-s + 1.92i·29-s + 1.24·31-s − 0.522·33-s − 1.70i·37-s − 0.554·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.049879322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.049879322\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 15iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.945334489453855508915989127609, −8.425524660694140207033237339247, −7.63265843011285257335009493275, −7.00304214269075922175671002007, −5.98089814944552536976952890433, −5.11884345635840674063961106964, −4.30987531222875992573776096024, −3.08521673887515311622077937819, −1.89303465965727662156161528727, −0.876262531421629958077602881430,
1.47973081456259457162912681181, 2.36693742157622901068982187610, 4.05116931983957084273570151249, 4.47945833697788903812045583287, 5.15638641956372212996978496981, 6.46067345777764791777940945302, 7.17353854376480542809786109343, 8.085073982003788169188022837109, 8.737073049638742642994390152970, 9.823566576200825298969625767688
