| L(s) = 1 | − 1.41i·3-s − 4.24·7-s + 0.999·9-s − 5.65i·11-s − 2i·13-s − 6·17-s − 2.82i·19-s + 6i·21-s + 7.07·23-s − 5.65i·27-s + 4i·29-s + 2.82·31-s − 8.00·33-s − 2i·37-s − 2.82·39-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s − 1.60·7-s + 0.333·9-s − 1.70i·11-s − 0.554i·13-s − 1.45·17-s − 0.648i·19-s + 1.30i·21-s + 1.47·23-s − 1.08i·27-s + 0.742i·29-s + 0.508·31-s − 1.39·33-s − 0.328i·37-s − 0.452·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4869316889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4869316889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 1.41iT - 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 2.82iT - 59T^{2} \) |
| 61 | \( 1 - 14iT - 61T^{2} \) |
| 67 | \( 1 - 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.357979040210222510042235318109, −7.06595783243019650090753817116, −6.85587097202252289816848869102, −6.13750876012166040545444477804, −5.34875685075280458305508789334, −4.15549126709171501446856606202, −3.14214245085775495247024182010, −2.66687396725846354100161100346, −1.11046800699204804839025176366, −0.16232167247119800372126578819,
1.75155587832882000337387975984, 2.82336084462008636173033952304, 3.77263282140169461112918312702, 4.46536479420125400518651568974, 5.05221487771172827511651040678, 6.38784479031823761813414918640, 6.77468514865452467429312961780, 7.41834551537185633111699430467, 8.682859984679428020459946166776, 9.352974750745270284735656756354
