L(s) = 1 | − 2.46·3-s + 3.32·5-s + 0.836·7-s + 3.06·9-s + 4.97·11-s − 6.31·13-s − 8.18·15-s − 17-s − 5.72·19-s − 2.06·21-s + 2.54·23-s + 6.04·25-s − 0.161·27-s + 4.47·29-s + 1.37·31-s − 12.2·33-s + 2.78·35-s − 6.12·37-s + 15.5·39-s − 0.359·41-s + 1.07·43-s + 10.1·45-s − 5.58·47-s − 6.30·49-s + 2.46·51-s − 1.86·53-s + 16.5·55-s + ⋯ |
L(s) = 1 | − 1.42·3-s + 1.48·5-s + 0.316·7-s + 1.02·9-s + 1.50·11-s − 1.75·13-s − 2.11·15-s − 0.242·17-s − 1.31·19-s − 0.449·21-s + 0.530·23-s + 1.20·25-s − 0.0310·27-s + 0.831·29-s + 0.247·31-s − 2.13·33-s + 0.469·35-s − 1.00·37-s + 2.49·39-s − 0.0561·41-s + 0.163·43-s + 1.51·45-s − 0.814·47-s − 0.900·49-s + 0.344·51-s − 0.256·53-s + 2.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 - 0.836T + 7T^{2} \) |
| 11 | \( 1 - 4.97T + 11T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 + 6.12T + 37T^{2} \) |
| 41 | \( 1 + 0.359T + 41T^{2} \) |
| 43 | \( 1 - 1.07T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + 1.86T + 53T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 3.91T + 73T^{2} \) |
| 79 | \( 1 + 5.94T + 79T^{2} \) |
| 83 | \( 1 + 3.90T + 83T^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−6.93849980523683251225075567864, −6.75160480122578721521094164536, −6.10479339026677391086858791221, −5.47140426836539217857429273993, −4.76736305169470983461855418889, −4.37210299419737081830037482386, −2.92064832943125096244599101912, −1.96866182400388693419849945232, −1.29920552817490245901324110762, 0,
1.29920552817490245901324110762, 1.96866182400388693419849945232, 2.92064832943125096244599101912, 4.37210299419737081830037482386, 4.76736305169470983461855418889, 5.47140426836539217857429273993, 6.10479339026677391086858791221, 6.75160480122578721521094164536, 6.93849980523683251225075567864