| L(s) = 1 | − 26.9·2-s + 111.·3-s + 216.·4-s + 2.12e3·5-s − 3.01e3·6-s + 8.47e3·7-s + 7.97e3·8-s − 7.21e3·9-s − 5.73e4·10-s + 7.76e3·11-s + 2.41e4·12-s + 1.88e5·13-s − 2.28e5·14-s + 2.37e5·15-s − 3.26e5·16-s + 1.94e5·18-s + 1.17e5·19-s + 4.60e5·20-s + 9.46e5·21-s − 2.09e5·22-s − 1.84e6·23-s + 8.90e5·24-s + 2.56e6·25-s − 5.09e6·26-s − 3.00e6·27-s + 1.83e6·28-s + 5.18e6·29-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.795·3-s + 0.422·4-s + 1.52·5-s − 0.949·6-s + 1.33·7-s + 0.688·8-s − 0.366·9-s − 1.81·10-s + 0.159·11-s + 0.336·12-s + 1.83·13-s − 1.59·14-s + 1.21·15-s − 1.24·16-s + 0.437·18-s + 0.207·19-s + 0.643·20-s + 1.06·21-s − 0.190·22-s − 1.37·23-s + 0.548·24-s + 1.31·25-s − 2.18·26-s − 1.08·27-s + 0.563·28-s + 1.36·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.991307182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.991307182\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 26.9T + 512T^{2} \) |
| 3 | \( 1 - 111.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.12e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.47e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.76e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.88e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 1.17e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.84e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.70e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.37e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.90e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.09e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.54e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.51e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.17e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.17e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.97e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.02e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.95e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.90e8T + 7.60e17T^{2} \) |
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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
−10.03100543163517832974311781553, −9.097111455647823350815690210879, −8.448844051830045660234519288855, −7.976642603266745814467599473957, −6.44091462639224781213467261646, −5.47423822317427574163497292351, −4.13269726935552020213744771097, −2.50372041899450313783280597158, −1.65335603306259668473293904245, −0.992492358000121115136001978629,
0.992492358000121115136001978629, 1.65335603306259668473293904245, 2.50372041899450313783280597158, 4.13269726935552020213744771097, 5.47423822317427574163497292351, 6.44091462639224781213467261646, 7.976642603266745814467599473957, 8.448844051830045660234519288855, 9.097111455647823350815690210879, 10.03100543163517832974311781553
