| L(s) = 1 | + 1.67e7·2-s − 1.33e11·3-s + 2.81e14·4-s + 1.87e17·5-s − 2.23e18·6-s + 8.28e20·7-s + 4.72e21·8-s − 2.21e23·9-s + 3.14e24·10-s − 3.05e25·11-s − 3.75e25·12-s − 4.92e25·13-s + 1.38e28·14-s − 2.50e28·15-s + 7.92e28·16-s + 2.50e30·17-s − 3.71e30·18-s + 5.26e30·19-s + 5.28e31·20-s − 1.10e32·21-s − 5.12e32·22-s − 7.99e32·23-s − 6.29e32·24-s + 1.74e34·25-s − 8.26e32·26-s + 6.14e34·27-s + 2.33e35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.272·3-s + 0.5·4-s + 1.40·5-s − 0.192·6-s + 1.63·7-s + 0.353·8-s − 0.925·9-s + 0.996·10-s − 0.935·11-s − 0.136·12-s − 0.0251·13-s + 1.15·14-s − 0.384·15-s + 0.250·16-s + 1.79·17-s − 0.654·18-s + 0.246·19-s + 0.704·20-s − 0.445·21-s − 0.661·22-s − 0.347·23-s − 0.0964·24-s + 0.984·25-s − 0.0178·26-s + 0.525·27-s + 0.817·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(\approx\) |
\(4.123845134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.123845134\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.67e7T \) |
| good | 3 | \( 1 + 1.33e11T + 2.39e23T^{2} \) |
| 5 | \( 1 - 1.87e17T + 1.77e34T^{2} \) |
| 7 | \( 1 - 8.28e20T + 2.56e41T^{2} \) |
| 11 | \( 1 + 3.05e25T + 1.06e51T^{2} \) |
| 13 | \( 1 + 4.92e25T + 3.83e54T^{2} \) |
| 17 | \( 1 - 2.50e30T + 1.95e60T^{2} \) |
| 19 | \( 1 - 5.26e30T + 4.55e62T^{2} \) |
| 23 | \( 1 + 7.99e32T + 5.30e66T^{2} \) |
| 29 | \( 1 + 2.91e35T + 4.54e71T^{2} \) |
| 31 | \( 1 - 5.98e36T + 1.19e73T^{2} \) |
| 37 | \( 1 - 1.31e38T + 6.94e76T^{2} \) |
| 41 | \( 1 + 2.57e39T + 1.06e79T^{2} \) |
| 43 | \( 1 - 3.01e39T + 1.09e80T^{2} \) |
| 47 | \( 1 + 1.35e41T + 8.56e81T^{2} \) |
| 53 | \( 1 - 3.38e42T + 3.08e84T^{2} \) |
| 59 | \( 1 + 1.39e42T + 5.91e86T^{2} \) |
| 61 | \( 1 + 8.43e43T + 3.02e87T^{2} \) |
| 67 | \( 1 - 4.28e44T + 3.00e89T^{2} \) |
| 71 | \( 1 + 1.65e45T + 5.14e90T^{2} \) |
| 73 | \( 1 + 5.25e45T + 2.00e91T^{2} \) |
| 79 | \( 1 - 1.45e46T + 9.63e92T^{2} \) |
| 83 | \( 1 + 9.24e46T + 1.08e94T^{2} \) |
| 89 | \( 1 - 8.22e46T + 3.31e95T^{2} \) |
| 97 | \( 1 + 4.76e48T + 2.24e97T^{2} \) |
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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
−16.94338572577514840230696520490, −14.63842697430683460631080616061, −13.69632637739654228276171914503, −11.78024926452517289789238985645, −10.28255603028618506949661735125, −8.029601410615503291712199546654, −5.78881220888448067201822684551, −5.05986559254949980047816221721, −2.65988708517971336327308455761, −1.35713215212954564094521439019,
1.35713215212954564094521439019, 2.65988708517971336327308455761, 5.05986559254949980047816221721, 5.78881220888448067201822684551, 8.029601410615503291712199546654, 10.28255603028618506949661735125, 11.78024926452517289789238985645, 13.69632637739654228276171914503, 14.63842697430683460631080616061, 16.94338572577514840230696520490
