| L(s) = 1 | + 3.23·3-s − 4.23·7-s + 7.47·9-s − 11-s + 2.23·13-s − 6.47·17-s − 19-s − 13.7·21-s − 23-s + 14.4·27-s − 1.76·29-s − 0.472·31-s − 3.23·33-s − 11.2·37-s + 7.23·39-s − 5.94·41-s + 2.52·43-s + 11.7·47-s + 10.9·49-s − 20.9·51-s − 1.23·53-s − 3.23·57-s − 3.23·59-s − 7.23·61-s − 31.6·63-s + 12.9·67-s − 3.23·69-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 1.60·7-s + 2.49·9-s − 0.301·11-s + 0.620·13-s − 1.56·17-s − 0.229·19-s − 2.99·21-s − 0.208·23-s + 2.78·27-s − 0.327·29-s − 0.0847·31-s − 0.563·33-s − 1.84·37-s + 1.15·39-s − 0.928·41-s + 0.385·43-s + 1.70·47-s + 1.56·49-s − 2.93·51-s − 0.169·53-s − 0.428·57-s − 0.421·59-s − 0.926·61-s − 3.98·63-s + 1.58·67-s − 0.389·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 9.47T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−7.38041808620477699267997240543, −6.81066338923647229295331314839, −6.34800328553629890845211848359, −5.24497203391693027152205599549, −4.05893958557440277932928343158, −3.82258086861460767264678641233, −2.95225363993380610756723987635, −2.46384713756264172515887997472, −1.57225848276104265504312742088, 0,
1.57225848276104265504312742088, 2.46384713756264172515887997472, 2.95225363993380610756723987635, 3.82258086861460767264678641233, 4.05893958557440277932928343158, 5.24497203391693027152205599549, 6.34800328553629890845211848359, 6.81066338923647229295331314839, 7.38041808620477699267997240543
