| L(s) = 1 | + 16·2-s − 13.9·3-s + 256·4-s − 1.71e3·5-s − 223.·6-s + 4.09e3·8-s − 1.94e4·9-s − 2.75e4·10-s − 7.13e4·11-s − 3.57e3·12-s + 1.56e5·13-s + 2.40e4·15-s + 6.55e4·16-s + 5.11e5·17-s − 3.11e5·18-s − 1.94e5·19-s − 4.40e5·20-s − 1.14e6·22-s − 1.08e5·23-s − 5.72e4·24-s + 1.00e6·25-s + 2.50e6·26-s + 5.47e5·27-s + 4.21e6·29-s + 3.84e5·30-s + 3.16e6·31-s + 1.04e6·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0996·3-s + 0.5·4-s − 1.22·5-s − 0.0704·6-s + 0.353·8-s − 0.990·9-s − 0.869·10-s − 1.46·11-s − 0.0498·12-s + 1.52·13-s + 0.122·15-s + 0.250·16-s + 1.48·17-s − 0.700·18-s − 0.342·19-s − 0.614·20-s − 1.03·22-s − 0.0805·23-s − 0.0352·24-s + 0.512·25-s + 1.07·26-s + 0.198·27-s + 1.10·29-s + 0.0866·30-s + 0.615·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.105256593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.105256593\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 13.9T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.71e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 7.13e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.56e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.11e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.94e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.08e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.21e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.16e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.44e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.69e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.64e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.52e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.78e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.14e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.01e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.08e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.84e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.84e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.64e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.57e8T + 7.60e17T^{2} \) |
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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
−12.04472019212043570613260893341, −11.28285758003141557509548246998, −10.35800175569439405351680058138, −8.360365372448191287376948953217, −7.82500159931142020157225795720, −6.21375575086957375847365788522, −5.13795254450904142136260240521, −3.75237955383198541478909339359, −2.80671812137656156278073960213, −0.72617702354398773897141345531,
0.72617702354398773897141345531, 2.80671812137656156278073960213, 3.75237955383198541478909339359, 5.13795254450904142136260240521, 6.21375575086957375847365788522, 7.82500159931142020157225795720, 8.360365372448191287376948953217, 10.35800175569439405351680058138, 11.28285758003141557509548246998, 12.04472019212043570613260893341
