| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−7.67100691982592494319962475267, −7.18213666900942543000580526143, −7.07665093602553087222345303659, −5.87723075347834717108006791312, −4.88361477114835320659974691443, −4.18130388667794124337835390943, −3.70967122958870468544864992828, −2.46040909099634949532930674713, −1.09743047650742575618685524963, 0,
1.32837498739201785889138248628, 2.60520460625860638316609689475, 3.53563637696182971539412512751, 4.36734315782402222249206983174, 5.39548168780312930976396048678, 5.72215104117866980872136179120, 6.55695265434507127195503621243, 7.47767137471417224354679673393, 8.407437464506144848470774426634
