| L(s) = 1 | + 2.82·3-s + 5.00·9-s − 2.82·11-s + 6·17-s − 8.48·19-s − 5·25-s + 5.65·27-s − 8.00·33-s + 6·41-s + 8.48·43-s − 7·49-s + 16.9·51-s − 24·57-s − 14.1·59-s − 8.48·67-s + 2·73-s − 14.1·75-s + 1.00·81-s + 2.82·83-s + 18·89-s − 10·97-s − 14.1·99-s + 19.7·107-s + 18·113-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 1.66·9-s − 0.852·11-s + 1.45·17-s − 1.94·19-s − 25-s + 1.08·27-s − 1.39·33-s + 0.937·41-s + 1.29·43-s − 49-s + 2.37·51-s − 3.17·57-s − 1.84·59-s − 1.03·67-s + 0.234·73-s − 1.63·75-s + 0.111·81-s + 0.310·83-s + 1.90·89-s − 1.01·97-s − 1.42·99-s + 1.91·107-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.956921643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956921643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−12.35797581564216331802133964919, −10.80930439699301989240452536246, −9.938036229146395142845664532292, −9.033934658635621728990784548147, −8.066335838706079171976710141123, −7.55299395754220445658708645745, −6.00140065324702785753059105988, −4.39774330912317846963186570778, −3.22834670546030511963807300880, −2.09259386172801032688627778191,
2.09259386172801032688627778191, 3.22834670546030511963807300880, 4.39774330912317846963186570778, 6.00140065324702785753059105988, 7.55299395754220445658708645745, 8.066335838706079171976710141123, 9.033934658635621728990784548147, 9.938036229146395142845664532292, 10.80930439699301989240452536246, 12.35797581564216331802133964919
