| L(s) = 1 | + 2-s + 2.61·3-s + 4-s − 1.23·5-s + 2.61·6-s − 2·7-s + 8-s + 3.85·9-s − 1.23·10-s + 2.61·12-s − 3.23·13-s − 2·14-s − 3.23·15-s + 16-s + 1.61·17-s + 3.85·18-s + 0.854·19-s − 1.23·20-s − 5.23·21-s − 3.23·23-s + 2.61·24-s − 3.47·25-s − 3.23·26-s + 2.23·27-s − 2·28-s − 4.47·29-s − 3.23·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s − 0.552·5-s + 1.06·6-s − 0.755·7-s + 0.353·8-s + 1.28·9-s − 0.390·10-s + 0.755·12-s − 0.897·13-s − 0.534·14-s − 0.835·15-s + 0.250·16-s + 0.392·17-s + 0.908·18-s + 0.195·19-s − 0.276·20-s − 1.14·21-s − 0.674·23-s + 0.534·24-s − 0.694·25-s − 0.634·26-s + 0.430·27-s − 0.377·28-s − 0.830·29-s − 0.590·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.405614830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.405614830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 + 0.0901T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 6.32T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−12.45618112546636037692280963740, −11.36292163946739514071372102626, −9.932749541309653048024732098464, −9.292889371651074355920669472020, −7.925476634561000018722407887335, −7.42742997818109701165262739129, −6.00283944129234321138458958830, −4.35330270706399664727065628455, −3.40998762431223147088630745347, −2.39534755043460532782145014653,
2.39534755043460532782145014653, 3.40998762431223147088630745347, 4.35330270706399664727065628455, 6.00283944129234321138458958830, 7.42742997818109701165262739129, 7.925476634561000018722407887335, 9.292889371651074355920669472020, 9.932749541309653048024732098464, 11.36292163946739514071372102626, 12.45618112546636037692280963740
