L(s) = 1 | + 2.15·3-s + 0.0591·5-s + 1.63·9-s − 5.19·11-s − 1.33·13-s + 0.127·15-s + 5.25·17-s + 6.42·19-s − 5.30·23-s − 4.99·25-s − 2.94·27-s − 2.05·29-s + 4.21·31-s − 11.1·33-s − 6.95·37-s − 2.87·39-s + 41-s − 4.70·43-s + 0.0964·45-s − 4.09·47-s + 11.3·51-s + 2.41·53-s − 0.307·55-s + 13.8·57-s + 4.59·59-s + 4.59·61-s − 0.0790·65-s + ⋯ |
L(s) = 1 | + 1.24·3-s + 0.0264·5-s + 0.543·9-s − 1.56·11-s − 0.370·13-s + 0.0328·15-s + 1.27·17-s + 1.47·19-s − 1.10·23-s − 0.999·25-s − 0.566·27-s − 0.382·29-s + 0.756·31-s − 1.94·33-s − 1.14·37-s − 0.460·39-s + 0.156·41-s − 0.717·43-s + 0.0143·45-s − 0.597·47-s + 1.58·51-s + 0.331·53-s − 0.0414·55-s + 1.83·57-s + 0.597·59-s + 0.588·61-s − 0.00980·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 - 0.0591T + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 1.33T + 13T^{2} \) |
| 17 | \( 1 - 5.25T + 17T^{2} \) |
| 19 | \( 1 - 6.42T + 19T^{2} \) |
| 23 | \( 1 + 5.30T + 23T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 4.09T + 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 - 4.59T + 61T^{2} \) |
| 67 | \( 1 + 9.74T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 0.670T + 89T^{2} \) |
| 97 | \( 1 - 3.02T + 97T^{2} \) |
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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
−7.47594665632227878278745090957, −7.32434799641919510202319805975, −5.84317931975976654799897860631, −5.51307890506739894842907559613, −4.62471575167709369427274169191, −3.58718700405558541670842264594, −3.09305146987166575147714675730, −2.40523097744330703097147545509, −1.52253867369232910440420346194, 0,
1.52253867369232910440420346194, 2.40523097744330703097147545509, 3.09305146987166575147714675730, 3.58718700405558541670842264594, 4.62471575167709369427274169191, 5.51307890506739894842907559613, 5.84317931975976654799897860631, 7.32434799641919510202319805975, 7.47594665632227878278745090957