| L(s) = 1 | + 2.36·2-s + 3.57·4-s − 3.86·7-s + 3.71·8-s + 2.15·11-s − 0.289·13-s − 9.11·14-s + 1.63·16-s + 6.14·17-s + 2.80·19-s + 5.09·22-s + 0.0934·23-s − 0.683·26-s − 13.8·28-s + 0.794·29-s − 2.43·31-s − 3.58·32-s + 14.4·34-s + 37-s + 6.63·38-s − 7.71·41-s + 1.72·43-s + 7.71·44-s + 0.220·46-s + 11.9·47-s + 7.90·49-s − 1.03·52-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.78·4-s − 1.45·7-s + 1.31·8-s + 0.650·11-s − 0.0803·13-s − 2.43·14-s + 0.407·16-s + 1.48·17-s + 0.644·19-s + 1.08·22-s + 0.0194·23-s − 0.134·26-s − 2.60·28-s + 0.147·29-s − 0.437·31-s − 0.634·32-s + 2.48·34-s + 0.164·37-s + 1.07·38-s − 1.20·41-s + 0.263·43-s + 1.16·44-s + 0.0325·46-s + 1.74·47-s + 1.12·49-s − 0.143·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.242582826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.242582826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 0.289T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 - 0.0934T + 23T^{2} \) |
| 29 | \( 1 - 0.794T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 41 | \( 1 + 7.71T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 4.34T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 9.27T + 67T^{2} \) |
| 71 | \( 1 + 6.28T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.25529682285382834298777714308, −7.00071235374621757341300585837, −6.18564263398653188922238766743, −5.66261189084709353322392036192, −5.13014173486782471247418960656, −4.05619107579027147482576794695, −3.57874992524861672646307824103, −3.06624553813617193579761944137, −2.20420269872328726169332585496, −0.869063144698149159943166507109,
0.869063144698149159943166507109, 2.20420269872328726169332585496, 3.06624553813617193579761944137, 3.57874992524861672646307824103, 4.05619107579027147482576794695, 5.13014173486782471247418960656, 5.66261189084709353322392036192, 6.18564263398653188922238766743, 7.00071235374621757341300585837, 7.25529682285382834298777714308
