| L(s) = 1 | + 3-s + 5-s + 0.772·7-s + 9-s + 3.49·11-s − 3.80·13-s + 15-s − 1.47·17-s + 5.60·19-s + 0.772·21-s + 4.75·23-s + 25-s + 27-s − 1.97·29-s − 0.171·31-s + 3.49·33-s + 0.772·35-s + 5.95·37-s − 3.80·39-s − 6.79·41-s + 2.90·43-s + 45-s + 0.666·47-s − 6.40·49-s − 1.47·51-s + 2.18·53-s + 3.49·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.291·7-s + 0.333·9-s + 1.05·11-s − 1.05·13-s + 0.258·15-s − 0.357·17-s + 1.28·19-s + 0.168·21-s + 0.990·23-s + 0.200·25-s + 0.192·27-s − 0.367·29-s − 0.0308·31-s + 0.608·33-s + 0.130·35-s + 0.978·37-s − 0.609·39-s − 1.06·41-s + 0.442·43-s + 0.149·45-s + 0.0972·47-s − 0.914·49-s − 0.206·51-s + 0.299·53-s + 0.470·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.957702008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957702008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 0.772T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 4.75T + 23T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 + 0.171T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 0.666T + 47T^{2} \) |
| 53 | \( 1 - 2.18T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 8.97T + 61T^{2} \) |
| 71 | \( 1 + 9.65T + 71T^{2} \) |
| 73 | \( 1 + 2.54T + 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 1.06T + 97T^{2} \) |
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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
−8.583279156379323306653679602715, −7.61103471395924611086273327905, −7.10586775906141583318430465714, −6.35657095414127712692451257926, −5.33798787888291592941223749352, −4.72618653937940659467759194246, −3.76823816926161184924745730374, −2.90577685248693583417071702984, −2.01822772513790315274726660076, −1.01479507601889089896201613233,
1.01479507601889089896201613233, 2.01822772513790315274726660076, 2.90577685248693583417071702984, 3.76823816926161184924745730374, 4.72618653937940659467759194246, 5.33798787888291592941223749352, 6.35657095414127712692451257926, 7.10586775906141583318430465714, 7.61103471395924611086273327905, 8.583279156379323306653679602715
