| L(s) = 1 | − i·3-s − 5-s + 3i·7-s + 2·9-s + 2·11-s + (3 − 2i)13-s + i·15-s + 3·17-s + 3·21-s + 6·23-s − 4·25-s − 5i·27-s − 6i·29-s − 2i·33-s − 3i·35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447·5-s + 1.13i·7-s + 0.666·9-s + 0.603·11-s + (0.832 − 0.554i)13-s + 0.258i·15-s + 0.727·17-s + 0.654·21-s + 1.25·23-s − 0.800·25-s − 0.962i·27-s − 1.11i·29-s − 0.348i·33-s − 0.507i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43781 - 0.142371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43781 - 0.142371i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−11.51934658035853730324972508152, −10.25192866628245335004599790864, −9.279794348719621932061108838309, −8.342221936283905436847609548821, −7.55506918318717107045340125087, −6.45057159331266784874117649187, −5.60103366058374838686862810103, −4.22324818397657834157527136281, −2.92416774946599500699904970660, −1.35309831046559301069422652713,
1.29470269968441608072715738381, 3.61102415332154168227078522061, 4.06480225066723033621650033340, 5.28237839942400213115685528030, 6.80857743730259293062239842643, 7.34864537020561679716376479234, 8.618797312506838930781956045142, 9.499410376287534354260725032362, 10.45332787133852351827811142108, 11.02002109623427657805421566937
