L(s) = 1 | + 2.32·3-s + 1.34·5-s + 2.42·9-s + 0.563·11-s − 4.61·13-s + 3.12·15-s − 1.38·17-s − 2.28·19-s − 23-s − 3.20·25-s − 1.33·27-s − 0.714·29-s − 7.27·31-s + 1.31·33-s + 3.33·37-s − 10.7·39-s − 5.66·41-s + 2.57·43-s + 3.25·45-s − 4.96·47-s − 3.23·51-s − 8.82·53-s + 0.754·55-s − 5.31·57-s − 5.96·59-s + 2.37·61-s − 6.18·65-s + ⋯ |
L(s) = 1 | + 1.34·3-s + 0.599·5-s + 0.808·9-s + 0.169·11-s − 1.28·13-s + 0.806·15-s − 0.336·17-s − 0.523·19-s − 0.208·23-s − 0.640·25-s − 0.256·27-s − 0.132·29-s − 1.30·31-s + 0.228·33-s + 0.547·37-s − 1.72·39-s − 0.884·41-s + 0.393·43-s + 0.485·45-s − 0.724·47-s − 0.452·51-s − 1.21·53-s + 0.101·55-s − 0.704·57-s − 0.777·59-s + 0.304·61-s − 0.767·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 - 0.563T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 29 | \( 1 + 0.714T + 29T^{2} \) |
| 31 | \( 1 + 7.27T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 + 5.66T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 + 5.96T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 - 0.900T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 - 4.50T + 97T^{2} \) |
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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
−7.54357682072595977588195911393, −6.86155238028528805864626383691, −6.07438630110208076893031569411, −5.27973504869689654959690228527, −4.48604842156399123740633505877, −3.71627759903729998584842915563, −2.94345314481427386707090468530, −2.18670385807499442238064860311, −1.71752333292358733297231862516, 0,
1.71752333292358733297231862516, 2.18670385807499442238064860311, 2.94345314481427386707090468530, 3.71627759903729998584842915563, 4.48604842156399123740633505877, 5.27973504869689654959690228527, 6.07438630110208076893031569411, 6.86155238028528805864626383691, 7.54357682072595977588195911393