| L(s) = 1 | − 2-s + 4-s + (−1.88 − 1.20i)5-s − 8-s + (1.88 + 1.20i)10-s + 1.59i·11-s + 0.925·13-s + 16-s − 3.95i·17-s − 0.625i·19-s + (−1.88 − 1.20i)20-s − 1.59i·22-s + 7.78·23-s + (2.08 + 4.54i)25-s − 0.925·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.841 − 0.539i)5-s − 0.353·8-s + (0.595 + 0.381i)10-s + 0.480i·11-s + 0.256·13-s + 0.250·16-s − 0.959i·17-s − 0.143i·19-s + (−0.420 − 0.269i)20-s − 0.339i·22-s + 1.62·23-s + (0.416 + 0.908i)25-s − 0.181·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003496057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003496057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.88 + 1.20i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 1.59iT - 11T^{2} \) |
| 13 | \( 1 - 0.925T + 13T^{2} \) |
| 17 | \( 1 + 3.95iT - 17T^{2} \) |
| 19 | \( 1 + 0.625iT - 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 + 9.34iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 0.426iT - 37T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 43 | \( 1 - 6.27iT - 43T^{2} \) |
| 47 | \( 1 - 2.78iT - 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 0.356iT - 67T^{2} \) |
| 71 | \( 1 - 9.07iT - 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 + 0.809iT - 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 - 7.87T + 97T^{2} \) |
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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
−8.313615094087308914496822566976, −7.60981708354069923375892936120, −7.01924630198410494980102768450, −6.32599667143641505887228719641, −5.06438485166353896531918045722, −4.72252757202977577886847653023, −3.53723599567618806622773490462, −2.80101827878506292454625223213, −1.52479183390533444269040880724, −0.54343483280042718524010465337,
0.75738401136725853509501513011, 1.96042199629122496061609465574, 3.14544118113477500898796682499, 3.61094584534305246111500836692, 4.68097338797898980127377462734, 5.68171195271936735724693079370, 6.48782944158451236683098837051, 7.13004676303084825023861719575, 7.74613005493346171265501313809, 8.547764422841170629646454298754
