| L(s) = 1 | + 1.73i·3-s − 1.73i·5-s + 6·9-s + 15·11-s + 13.8i·13-s + 2.99·15-s + 29.4i·17-s − 15.5i·19-s − 9·23-s + 22·25-s + 25.9i·27-s − 6·29-s − 12.1i·31-s + 25.9i·33-s + 31·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.346i·5-s + 0.666·9-s + 1.36·11-s + 1.06i·13-s + 0.199·15-s + 1.73i·17-s − 0.820i·19-s − 0.391·23-s + 0.880·25-s + 0.962i·27-s − 0.206·29-s − 0.391i·31-s + 0.787i·33-s + 0.837·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.55900 + 0.581234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55900 + 0.581234i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.73iT - 9T^{2} \) |
| 5 | \( 1 + 1.73iT - 25T^{2} \) |
| 11 | \( 1 - 15T + 121T^{2} \) |
| 13 | \( 1 - 13.8iT - 169T^{2} \) |
| 17 | \( 1 - 29.4iT - 289T^{2} \) |
| 19 | \( 1 + 15.5iT - 361T^{2} \) |
| 23 | \( 1 + 9T + 529T^{2} \) |
| 29 | \( 1 + 6T + 841T^{2} \) |
| 31 | \( 1 + 12.1iT - 961T^{2} \) |
| 37 | \( 1 - 31T + 1.36e3T^{2} \) |
| 41 | \( 1 + 55.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 57T + 2.80e3T^{2} \) |
| 59 | \( 1 + 81.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 49T + 4.48e3T^{2} \) |
| 71 | \( 1 + 126T + 5.04e3T^{2} \) |
| 73 | \( 1 + 25.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 73T + 6.24e3T^{2} \) |
| 83 | \( 1 + 13.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 57.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 27.7iT - 9.40e3T^{2} \) |
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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
−12.35926775789176139925422240420, −11.37455131204586835686804664314, −10.37217131407535820391687063059, −9.321886008618786069897424481340, −8.687569855091578378158742838403, −7.14246516245837854318538640629, −6.14872166303318715681070623214, −4.54654225404557146777243159871, −3.84152602179040602064574599742, −1.61235806314308795514742275512,
1.19441660867917543747394249423, 3.01198596002197248607982491459, 4.52132890865353843653829053775, 6.06811177587239410804549843682, 7.03770225506408392566435076058, 7.88775229080578425903758318121, 9.258554990191617181230622903350, 10.13153755679083717284731304266, 11.33899514230659472486388704876, 12.20242413000535838459649093154
