L(s) = 1 | + 3.14·3-s − 1.20·7-s + 6.86·9-s − 3.65·11-s − 0.859·13-s − 6.72·17-s + 1.51·19-s − 3.78·21-s − 23-s + 12.1·27-s + 0.548·29-s + 5.99·31-s − 11.4·33-s − 2.04·37-s − 2.69·39-s − 7.14·41-s − 10.0·43-s − 9.17·47-s − 5.54·49-s − 21.1·51-s − 5.37·53-s + 4.76·57-s + 0.582·59-s − 8.83·61-s − 8.28·63-s − 3.20·67-s − 3.14·69-s + ⋯ |
L(s) = 1 | + 1.81·3-s − 0.456·7-s + 2.28·9-s − 1.10·11-s − 0.238·13-s − 1.63·17-s + 0.348·19-s − 0.826·21-s − 0.208·23-s + 2.33·27-s + 0.101·29-s + 1.07·31-s − 1.99·33-s − 0.335·37-s − 0.432·39-s − 1.11·41-s − 1.53·43-s − 1.33·47-s − 0.792·49-s − 2.95·51-s − 0.738·53-s + 0.631·57-s + 0.0758·59-s − 1.13·61-s − 1.04·63-s − 0.391·67-s − 0.378·69-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 | \( 1 - 3.14T + 3T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 + 0.859T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 29 | \( 1 - 0.548T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 9.17T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 0.582T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 + 0.0700T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 - 4.96T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.54616101939108340775766664141, −6.83724378254979636308279181489, −6.30233177876461565577166675955, −4.94632553188118217973498448743, −4.59254676685944530530131877551, −3.50896585521808108279141877191, −3.05673984847327994104480725788, −2.33004802615365978329871545745, −1.65528332024576492738356464652, 0,
1.65528332024576492738356464652, 2.33004802615365978329871545745, 3.05673984847327994104480725788, 3.50896585521808108279141877191, 4.59254676685944530530131877551, 4.94632553188118217973498448743, 6.30233177876461565577166675955, 6.83724378254979636308279181489, 7.54616101939108340775766664141