| L(s) = 1 | − 1.42·2-s + 0.0371·4-s + 2.54·5-s − 7-s + 2.80·8-s − 3.62·10-s + 6.28·11-s + 2.89·13-s + 1.42·14-s − 4.07·16-s + 5.09·17-s − 4.74·19-s + 0.0944·20-s − 8.96·22-s − 5.85·23-s + 1.45·25-s − 4.13·26-s − 0.0371·28-s + 0.225·29-s + 3.19·31-s + 0.210·32-s − 7.26·34-s − 2.54·35-s − 6.42·37-s + 6.76·38-s + 7.11·40-s + 7.86·41-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.0185·4-s + 1.13·5-s − 0.377·7-s + 0.990·8-s − 1.14·10-s + 1.89·11-s + 0.803·13-s + 0.381·14-s − 1.01·16-s + 1.23·17-s − 1.08·19-s + 0.0211·20-s − 1.91·22-s − 1.22·23-s + 0.290·25-s − 0.810·26-s − 0.00702·28-s + 0.0418·29-s + 0.573·31-s + 0.0371·32-s − 1.24·34-s − 0.429·35-s − 1.05·37-s + 1.09·38-s + 1.12·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.636815846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.636815846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.42T + 2T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 - 6.28T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 - 0.225T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 - 7.86T + 41T^{2} \) |
| 43 | \( 1 + 4.35T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 - 3.98T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 5.28T + 79T^{2} \) |
| 83 | \( 1 - 8.77T + 83T^{2} \) |
| 89 | \( 1 + 9.56T + 89T^{2} \) |
| 97 | \( 1 - 0.248T + 97T^{2} \) |
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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
−8.070141935144609693259847750455, −7.16913232119686635703454608986, −6.36574010010819529932518992836, −6.08721399988984611926814522683, −5.17497500996548612925408279474, −4.04241818354070755293630531166, −3.70595276694099448531518151196, −2.26124784654574777805613860391, −1.54753328408784019944722444524, −0.815048748818792132525995495282,
0.815048748818792132525995495282, 1.54753328408784019944722444524, 2.26124784654574777805613860391, 3.70595276694099448531518151196, 4.04241818354070755293630531166, 5.17497500996548612925408279474, 6.08721399988984611926814522683, 6.36574010010819529932518992836, 7.16913232119686635703454608986, 8.070141935144609693259847750455
