| L(s) = 1 | + 4·11-s − 4·13-s + 6·17-s + 4·19-s − 4·23-s + 4·29-s + 4·37-s + 8·41-s + 12·47-s − 2·53-s + 12·59-s − 2·61-s + 8·67-s − 8·71-s + 16·73-s − 8·79-s + 8·83-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 0.742·29-s + 0.657·37-s + 1.24·41-s + 1.75·47-s − 0.274·53-s + 1.56·59-s − 0.256·61-s + 0.977·67-s − 0.949·71-s + 1.87·73-s − 0.900·79-s + 0.878·83-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.426801532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.426801532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.15272622877195, −13.50965597060653, −12.71121991834998, −12.36243251290976, −11.87293947337250, −11.68199899660931, −10.92437496596917, −10.30325317337783, −9.822220246135262, −9.482491496732896, −9.033737097585477, −8.234969250542823, −7.816404795861668, −7.299452076346895, −6.844042747351338, −6.134384203604341, −5.656434513629919, −5.157379820992318, −4.413002162383888, −3.942815930747850, −3.331167470005856, −2.651053909554837, −2.046667523371038, −1.103761832505936, −0.6989682652100675,
0.6989682652100675, 1.103761832505936, 2.046667523371038, 2.651053909554837, 3.331167470005856, 3.942815930747850, 4.413002162383888, 5.157379820992318, 5.656434513629919, 6.134384203604341, 6.844042747351338, 7.299452076346895, 7.816404795861668, 8.234969250542823, 9.033737097585477, 9.482491496732896, 9.822220246135262, 10.30325317337783, 10.92437496596917, 11.68199899660931, 11.87293947337250, 12.36243251290976, 12.71121991834998, 13.50965597060653, 14.15272622877195
