| L(s) = 1 | + 1.58·3-s + 2.84·7-s − 0.493·9-s − 1.98·11-s + 4.69·13-s + 3.16·17-s + 6.16·19-s + 4.50·21-s − 23-s − 5.53·27-s − 6.61·29-s + 8.29·31-s − 3.14·33-s − 1.71·37-s + 7.42·39-s + 6.72·41-s + 0.177·43-s − 11.4·47-s + 1.10·49-s + 5.01·51-s + 6.18·53-s + 9.75·57-s + 7.61·59-s + 11.8·61-s − 1.40·63-s − 3.07·67-s − 1.58·69-s + ⋯ |
| L(s) = 1 | + 0.914·3-s + 1.07·7-s − 0.164·9-s − 0.598·11-s + 1.30·13-s + 0.768·17-s + 1.41·19-s + 0.983·21-s − 0.208·23-s − 1.06·27-s − 1.22·29-s + 1.49·31-s − 0.547·33-s − 0.281·37-s + 1.18·39-s + 1.05·41-s + 0.0271·43-s − 1.67·47-s + 0.157·49-s + 0.702·51-s + 0.849·53-s + 1.29·57-s + 0.991·59-s + 1.51·61-s − 0.176·63-s − 0.375·67-s − 0.190·69-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
| \(L(1)\) |
\(\approx\) |
\(3.690225700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.690225700\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.58T + 3T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 6.16T + 19T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 + 1.71T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 0.177T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 3.07T + 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 + 3.49T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 7.00T + 97T^{2} \) |
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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
−7.81275533062113652639396639384, −7.39669954172911159674698222310, −6.29557120214045494242116713052, −5.52929469534979614651020603824, −5.08216677488891310996970563657, −4.01398048408968804058778725717, −3.41133082683224528720292833422, −2.68296292652576523796146526101, −1.78265457984193183639139460061, −0.934103500596506240602860314984,
0.934103500596506240602860314984, 1.78265457984193183639139460061, 2.68296292652576523796146526101, 3.41133082683224528720292833422, 4.01398048408968804058778725717, 5.08216677488891310996970563657, 5.52929469534979614651020603824, 6.29557120214045494242116713052, 7.39669954172911159674698222310, 7.81275533062113652639396639384
