| L(s) = 1 | + 3.27i·3-s − i·5-s + 1.27i·7-s − 7.71·9-s + 5.43i·11-s − 0.166·13-s + 3.27·15-s + (2.27 − 3.43i)17-s − 2.54·19-s − 4.16·21-s − 3.27i·23-s − 25-s − 15.4i·27-s + 6.54i·29-s + 5.98i·31-s + ⋯ |
| L(s) = 1 | + 1.88i·3-s − 0.447i·5-s + 0.481i·7-s − 2.57·9-s + 1.64i·11-s − 0.0462·13-s + 0.845·15-s + (0.551 − 0.834i)17-s − 0.584·19-s − 0.909·21-s − 0.682i·23-s − 0.200·25-s − 2.96i·27-s + 1.21i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.323026 + 1.07478i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.323026 + 1.07478i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 17 | \( 1 + (-2.27 + 3.43i)T \) |
| good | 3 | \( 1 - 3.27iT - 3T^{2} \) |
| 7 | \( 1 - 1.27iT - 7T^{2} \) |
| 11 | \( 1 - 5.43iT - 11T^{2} \) |
| 13 | \( 1 + 0.166T + 13T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 23 | \( 1 + 3.27iT - 23T^{2} \) |
| 29 | \( 1 - 6.54iT - 29T^{2} \) |
| 31 | \( 1 - 5.98iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 8.54iT - 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 - 7.42iT - 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 - 2.89iT - 71T^{2} \) |
| 73 | \( 1 - 12.5iT - 73T^{2} \) |
| 79 | \( 1 + 3.77iT - 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.83T + 89T^{2} \) |
| 97 | \( 1 - 1.45iT - 97T^{2} \) |
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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
−11.92713928588924861951507401186, −10.60299196454272440091379659376, −10.14433766882044252927766415985, −9.142413554524834773675483429568, −8.705718729382153773103449578657, −7.17773966355864231748480703975, −5.57034853120738956064666149108, −4.85140754377587187534291337806, −4.03663131742037353750634032180, −2.61262637833418317686123255019,
0.796622317057322197824446259341, 2.33833060985304386448120705085, 3.61801552685630505519917764559, 5.88229303880801374537888399171, 6.20682993424848811823532266776, 7.47790116703814081163643270956, 7.981279272462294912430743606766, 8.963415723500026764271642242215, 10.56117646586074915351476157175, 11.35275328273379455197218308190
