| L(s) = 1 | + 2.39·3-s − 1.16·5-s + 7-s + 2.73·9-s − 3.38·11-s − 2.57·13-s − 2.78·15-s − 17-s − 6.30·19-s + 2.39·21-s + 8.41·23-s − 3.65·25-s − 0.623·27-s − 5.38·29-s − 2.05·31-s − 8.11·33-s − 1.16·35-s + 3.25·37-s − 6.17·39-s + 7.90·41-s − 8.38·43-s − 3.18·45-s − 4.48·47-s + 49-s − 2.39·51-s − 1.33·53-s + 3.93·55-s + ⋯ |
| L(s) = 1 | + 1.38·3-s − 0.519·5-s + 0.377·7-s + 0.913·9-s − 1.02·11-s − 0.715·13-s − 0.718·15-s − 0.242·17-s − 1.44·19-s + 0.522·21-s + 1.75·23-s − 0.730·25-s − 0.120·27-s − 1.00·29-s − 0.368·31-s − 1.41·33-s − 0.196·35-s + 0.535·37-s − 0.989·39-s + 1.23·41-s − 1.27·43-s − 0.474·45-s − 0.653·47-s + 0.142·49-s − 0.335·51-s − 0.183·53-s + 0.530·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 + 2.57T + 13T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 - 8.41T + 23T^{2} \) |
| 29 | \( 1 + 5.38T + 29T^{2} \) |
| 31 | \( 1 + 2.05T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 - 7.90T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 1.33T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 1.13T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 7.74T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−8.163398167509702410955962098920, −7.56678674046580422237458714593, −7.03854207800228249099407054657, −5.85929196677025551532587867257, −4.87408974029167717340289542401, −4.22666677029096058284846617511, −3.27348743740102993803666407799, −2.57430036359983230931619612475, −1.79856758786627696526464639547, 0,
1.79856758786627696526464639547, 2.57430036359983230931619612475, 3.27348743740102993803666407799, 4.22666677029096058284846617511, 4.87408974029167717340289542401, 5.85929196677025551532587867257, 7.03854207800228249099407054657, 7.56678674046580422237458714593, 8.163398167509702410955962098920
