| L(s) = 1 | − 3-s + (2 + i)5-s − 4i·7-s + 9-s − 4i·11-s + (−2 − i)15-s + 4i·17-s + 4i·21-s + 4i·23-s + (3 + 4i)25-s − 27-s − 6i·29-s + 4·31-s + 4i·33-s + (4 − 8i)35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.894 + 0.447i)5-s − 1.51i·7-s + 0.333·9-s − 1.20i·11-s + (−0.516 − 0.258i)15-s + 0.970i·17-s + 0.872i·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s − 0.192·27-s − 1.11i·29-s + 0.718·31-s + 0.696i·33-s + (0.676 − 1.35i)35-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
| \(L(1)\) |
\(\approx\) |
\(1.774280956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.774280956\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.100339686624105484240114584092, −7.63731906834285617515929822828, −6.65894874970356530470862956954, −6.15116493275716350907595604770, −5.58998272192695646335070045087, −4.46442796500013484512843665417, −3.78725566099458640682077395931, −2.84455365631458794571511144179, −1.54984243871760249027867512690, −0.63134009277517368460804155646,
1.13379911430019667683309605047, 2.27108252912912709956622885863, 2.76640496182648206946222259235, 4.48843092112184210901224627671, 4.88304276254778416050683265833, 5.70920482603491429739545368637, 6.21814891903146690481519422026, 7.01765088310070113605281841117, 7.929210961192851874185850441225, 8.849374337804467343214093047852
