| L(s) = 1 | − 0.558·2-s + 0.670·3-s − 1.68·4-s − 0.374·6-s + 2.55·7-s + 2.06·8-s − 2.55·9-s − 4.90·11-s − 1.13·12-s − 4.22·13-s − 1.42·14-s + 2.22·16-s − 5.54·17-s + 1.42·18-s + 1.71·21-s + 2.74·22-s + 5.61·23-s + 1.38·24-s + 2.36·26-s − 3.72·27-s − 4.31·28-s + 6.58·29-s − 6.07·31-s − 5.36·32-s − 3.28·33-s + 3.09·34-s + 4.30·36-s + ⋯ |
| L(s) = 1 | − 0.394·2-s + 0.386·3-s − 0.843·4-s − 0.152·6-s + 0.965·7-s + 0.728·8-s − 0.850·9-s − 1.47·11-s − 0.326·12-s − 1.17·13-s − 0.381·14-s + 0.556·16-s − 1.34·17-s + 0.335·18-s + 0.373·21-s + 0.584·22-s + 1.17·23-s + 0.281·24-s + 0.463·26-s − 0.715·27-s − 0.815·28-s + 1.22·29-s − 1.09·31-s − 0.948·32-s − 0.572·33-s + 0.531·34-s + 0.717·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6610708315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6610708315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.558T + 2T^{2} \) |
| 3 | \( 1 - 0.670T + 3T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 + 4.22T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 23 | \( 1 - 5.61T + 23T^{2} \) |
| 29 | \( 1 - 6.58T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 - 0.883T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 1.37T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 2.30T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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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
−7.83364617664432396172848546744, −7.43966077321182455138021193636, −6.45075372926560025484472559504, −5.25317432898121587606944280956, −5.04668185147106880333993889861, −4.48498699991587752009613360453, −3.31970353262437945342894395566, −2.55925339078680942534206309993, −1.81035249914276458543325487225, −0.39496760281530219220235035939,
0.39496760281530219220235035939, 1.81035249914276458543325487225, 2.55925339078680942534206309993, 3.31970353262437945342894395566, 4.48498699991587752009613360453, 5.04668185147106880333993889861, 5.25317432898121587606944280956, 6.45075372926560025484472559504, 7.43966077321182455138021193636, 7.83364617664432396172848546744
