| L(s) = 1 | + 3-s + (1 + 2i)5-s + 9-s + 4i·11-s + 2·13-s + (1 + 2i)15-s − 2i·17-s − 2i·19-s − 6i·23-s + (−3 + 4i)25-s + 27-s + 8i·29-s + 8·31-s + 4i·33-s + 6·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (0.447 + 0.894i)5-s + 0.333·9-s + 1.20i·11-s + 0.554·13-s + (0.258 + 0.516i)15-s − 0.485i·17-s − 0.458i·19-s − 1.25i·23-s + (−0.600 + 0.800i)25-s + 0.192·27-s + 1.48i·29-s + 1.43·31-s + 0.696i·33-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.659212618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659212618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.700382686020305581566614467706, −7.86250809047460709688331463688, −6.97277078847425855413410134123, −6.74952239572403085484279716981, −5.72884876311585925854257164913, −4.73537110943838174246563630263, −4.03028866975095719117720014891, −2.87159732436951834497185665119, −2.45831178234769708689973395773, −1.28157948058322940001445269669,
0.76668829466505336779209408003, 1.71410433396475413249449058911, 2.78219898644770170930627616998, 3.75945022253446010858029934728, 4.37332572668820143007912395186, 5.58968782373938515454911508644, 5.88224850424786620632070811355, 6.84296105568389350330122129598, 8.007644658493753919732255863642, 8.320907708272620910301135254215
