| L(s) = 1 | + 2.72·3-s − 5-s + 5.23·7-s + 4.45·9-s + 0.867·11-s − 2.05·13-s − 2.72·15-s + 6.70·17-s − 0.0126·19-s + 14.2·21-s + 5.84·23-s + 25-s + 3.96·27-s − 29-s + 4.35·31-s + 2.36·33-s − 5.23·35-s − 11.6·37-s − 5.59·39-s − 0.467·41-s + 4.22·43-s − 4.45·45-s − 8.59·47-s + 20.3·49-s + 18.2·51-s − 13.8·53-s − 0.867·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 0.447·5-s + 1.97·7-s + 1.48·9-s + 0.261·11-s − 0.568·13-s − 0.704·15-s + 1.62·17-s − 0.00290·19-s + 3.11·21-s + 1.21·23-s + 0.200·25-s + 0.762·27-s − 0.185·29-s + 0.781·31-s + 0.412·33-s − 0.884·35-s − 1.91·37-s − 0.896·39-s − 0.0730·41-s + 0.644·43-s − 0.663·45-s − 1.25·47-s + 2.90·49-s + 2.56·51-s − 1.90·53-s − 0.116·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.156217746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.156217746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.72T + 3T^{2} \) |
| 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 - 0.867T + 11T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 + 0.0126T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 0.467T + 41T^{2} \) |
| 43 | \( 1 - 4.22T + 43T^{2} \) |
| 47 | \( 1 + 8.59T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 - 5.47T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 + 2.66T + 73T^{2} \) |
| 79 | \( 1 - 6.72T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 1.89T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.80143088864930956622141808633, −7.46922134229486279177636862060, −6.64290603566529560802727898198, −5.19022045416783674753478470447, −5.03330956498478227920193739903, −4.05241390185021450261994866169, −3.41278187899009618982399000670, −2.65331350260631340871820005670, −1.76535727072819974947632481771, −1.13462601515905021661946032442,
1.13462601515905021661946032442, 1.76535727072819974947632481771, 2.65331350260631340871820005670, 3.41278187899009618982399000670, 4.05241390185021450261994866169, 5.03330956498478227920193739903, 5.19022045416783674753478470447, 6.64290603566529560802727898198, 7.46922134229486279177636862060, 7.80143088864930956622141808633
