| L(s) = 1 | − 3.58e14·2-s − 7.12e22·3-s + 8.86e28·4-s − 1.64e33·5-s + 2.55e37·6-s − 1.35e40·7-s − 1.75e43·8-s + 2.96e45·9-s + 5.88e47·10-s + 4.57e49·11-s − 6.32e51·12-s + 7.91e52·13-s + 4.86e54·14-s + 1.17e56·15-s + 2.78e57·16-s − 3.60e57·17-s − 1.06e60·18-s − 1.15e60·19-s − 1.45e62·20-s + 9.68e62·21-s − 1.63e64·22-s − 4.43e64·23-s + 1.25e66·24-s + 1.79e65·25-s − 2.83e67·26-s − 5.99e67·27-s − 1.20e69·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.54·3-s + 2.23·4-s − 1.03·5-s + 2.78·6-s − 0.979·7-s − 2.22·8-s + 1.39·9-s + 1.86·10-s + 1.56·11-s − 3.46·12-s + 0.968·13-s + 1.76·14-s + 1.60·15-s + 1.77·16-s − 0.129·17-s − 2.51·18-s − 0.210·19-s − 2.31·20-s + 1.51·21-s − 2.81·22-s − 0.921·23-s + 3.45·24-s + 0.0709·25-s − 1.74·26-s − 0.613·27-s − 2.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(96-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+95/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(48)\) |
\(\approx\) |
\(0.1716360069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1716360069\) |
| \(L(\frac{97}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 3.58e14T + 3.96e28T^{2} \) |
| 3 | \( 1 + 7.12e22T + 2.12e45T^{2} \) |
| 5 | \( 1 + 1.64e33T + 2.52e66T^{2} \) |
| 7 | \( 1 + 1.35e40T + 1.92e80T^{2} \) |
| 11 | \( 1 - 4.57e49T + 8.55e98T^{2} \) |
| 13 | \( 1 - 7.91e52T + 6.67e105T^{2} \) |
| 17 | \( 1 + 3.60e57T + 7.80e116T^{2} \) |
| 19 | \( 1 + 1.15e60T + 3.03e121T^{2} \) |
| 23 | \( 1 + 4.43e64T + 2.31e129T^{2} \) |
| 29 | \( 1 - 3.11e69T + 8.46e138T^{2} \) |
| 31 | \( 1 + 6.79e70T + 4.77e141T^{2} \) |
| 37 | \( 1 + 2.19e74T + 9.53e148T^{2} \) |
| 41 | \( 1 + 7.21e76T + 1.63e153T^{2} \) |
| 43 | \( 1 - 9.95e76T + 1.51e155T^{2} \) |
| 47 | \( 1 - 1.61e79T + 7.06e158T^{2} \) |
| 53 | \( 1 + 2.46e81T + 6.40e163T^{2} \) |
| 59 | \( 1 - 1.31e84T + 1.70e168T^{2} \) |
| 61 | \( 1 - 8.03e84T + 4.03e169T^{2} \) |
| 67 | \( 1 + 4.90e86T + 2.99e173T^{2} \) |
| 71 | \( 1 - 4.39e87T + 7.40e175T^{2} \) |
| 73 | \( 1 + 3.36e88T + 1.03e177T^{2} \) |
| 79 | \( 1 - 1.11e89T + 1.88e180T^{2} \) |
| 83 | \( 1 + 1.87e91T + 2.05e182T^{2} \) |
| 89 | \( 1 - 3.70e92T + 1.55e185T^{2} \) |
| 97 | \( 1 + 2.62e94T + 5.53e188T^{2} \) |
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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
−15.79648860084458894381158016643, −12.08342735828055641368263814436, −11.35319699159688984743371290665, −10.12451440420336627795041106527, −8.659161271198330280717851376829, −6.95234684793055955288634814830, −6.21199111859026605985627503476, −3.77985058902174262790835553550, −1.38233507850864689515384922457, −0.35156161241683098745572636613,
0.35156161241683098745572636613, 1.38233507850864689515384922457, 3.77985058902174262790835553550, 6.21199111859026605985627503476, 6.95234684793055955288634814830, 8.659161271198330280717851376829, 10.12451440420336627795041106527, 11.35319699159688984743371290665, 12.08342735828055641368263814436, 15.79648860084458894381158016643
