| L(s) = 1 | − 38.2·2-s − 153.·3-s + 949.·4-s + 2.10e3·5-s + 5.87e3·6-s + 5.20e3·7-s − 1.67e4·8-s + 3.92e3·9-s − 8.05e4·10-s + 1.44e4·11-s − 1.45e5·12-s − 2.71e4·13-s − 1.98e5·14-s − 3.23e5·15-s + 1.52e5·16-s + 3.05e5·17-s − 1.49e5·18-s + 7.50e5·19-s + 2.00e6·20-s − 7.99e5·21-s − 5.52e5·22-s + 3.03e4·23-s + 2.56e6·24-s + 2.48e6·25-s + 1.03e6·26-s + 2.42e6·27-s + 4.93e6·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 1.09·3-s + 1.85·4-s + 1.50·5-s + 1.85·6-s + 0.819·7-s − 1.44·8-s + 0.199·9-s − 2.54·10-s + 0.297·11-s − 2.03·12-s − 0.263·13-s − 1.38·14-s − 1.65·15-s + 0.583·16-s + 0.888·17-s − 0.336·18-s + 1.32·19-s + 2.79·20-s − 0.896·21-s − 0.503·22-s + 0.0225·23-s + 1.58·24-s + 1.27·25-s + 0.445·26-s + 0.876·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.134209147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134209147\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 1.50e9T \) |
| good | 2 | \( 1 + 38.2T + 512T^{2} \) |
| 3 | \( 1 + 153.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.10e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.20e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.71e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.05e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.50e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.03e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.93e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.52e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.67e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.05e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.50e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 7.59e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.98e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.81e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.19e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.06e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.45e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.97e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.21e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.64e7T + 7.60e17T^{2} \) |
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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
−10.53485288718863756076602239266, −9.906875327945996334639183515156, −9.086275465481929966949259431640, −7.959274767613300191219665281936, −6.84354541263099422224873589244, −5.87202173746900331699292340674, −5.05743735881519411183394052606, −2.58383888565499948069171532781, −1.39557814240478518633408228100, −0.802263198127131588585034692665,
0.802263198127131588585034692665, 1.39557814240478518633408228100, 2.58383888565499948069171532781, 5.05743735881519411183394052606, 5.87202173746900331699292340674, 6.84354541263099422224873589244, 7.959274767613300191219665281936, 9.086275465481929966949259431640, 9.906875327945996334639183515156, 10.53485288718863756076602239266
