| L(s) = 1 | + 3.85·7-s + 4.85·11-s − 5.85·13-s − 7.85·17-s + 2·19-s − 1.85·23-s + 9.70·29-s − 10.7·31-s + 0.854·37-s − 8.56·41-s − 5.85·43-s − 6.70·47-s + 7.85·49-s + 1.85·53-s − 7.85·59-s − 5.85·61-s − 7·67-s − 9·71-s − 10.7·73-s + 18.7·77-s − 1.70·79-s + 6.70·83-s + 12·89-s − 22.5·91-s − 10·97-s + 1.85·101-s + 6.85·103-s + ⋯ |
| L(s) = 1 | + 1.45·7-s + 1.46·11-s − 1.62·13-s − 1.90·17-s + 0.458·19-s − 0.386·23-s + 1.80·29-s − 1.92·31-s + 0.140·37-s − 1.33·41-s − 0.892·43-s − 0.978·47-s + 1.12·49-s + 0.254·53-s − 1.02·59-s − 0.749·61-s − 0.855·67-s − 1.06·71-s − 1.25·73-s + 2.13·77-s − 0.192·79-s + 0.736·83-s + 1.27·89-s − 2.36·91-s − 1.01·97-s + 0.184·101-s + 0.675·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 0.854T + 37T^{2} \) |
| 41 | \( 1 + 8.56T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.41244401095218241510295828520, −6.87422441239768366869717959718, −6.22368863113951206636290610991, −5.11606487029049109194830953413, −4.69212994732545970290370627731, −4.15264310473703626780455690062, −3.03921043223235942382647659503, −1.99656971868463561027860322453, −1.51376062726110985846512672776, 0,
1.51376062726110985846512672776, 1.99656971868463561027860322453, 3.03921043223235942382647659503, 4.15264310473703626780455690062, 4.69212994732545970290370627731, 5.11606487029049109194830953413, 6.22368863113951206636290610991, 6.87422441239768366869717959718, 7.41244401095218241510295828520
