| L(s) = 1 | + 2-s + 0.767·3-s + 4-s + 2.24·5-s + 0.767·6-s + 7-s + 8-s − 2.41·9-s + 2.24·10-s − 4.92·11-s + 0.767·12-s + 2.23·13-s + 14-s + 1.71·15-s + 16-s + 2.78·17-s − 2.41·18-s + 2.24·20-s + 0.767·21-s − 4.92·22-s + 7.55·23-s + 0.767·24-s + 0.0213·25-s + 2.23·26-s − 4.15·27-s + 28-s + 4.83·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.442·3-s + 0.5·4-s + 1.00·5-s + 0.313·6-s + 0.377·7-s + 0.353·8-s − 0.803·9-s + 0.708·10-s − 1.48·11-s + 0.221·12-s + 0.618·13-s + 0.267·14-s + 0.443·15-s + 0.250·16-s + 0.674·17-s − 0.568·18-s + 0.501·20-s + 0.167·21-s − 1.04·22-s + 1.57·23-s + 0.156·24-s + 0.00426·25-s + 0.437·26-s − 0.798·27-s + 0.188·28-s + 0.896·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.476423302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.476423302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.767T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 - 9.15T + 31T^{2} \) |
| 37 | \( 1 + 0.687T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 9.26T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 - 8.72T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 + 3.82T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 2.39T + 89T^{2} \) |
| 97 | \( 1 + 5.96T + 97T^{2} \) |
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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
−8.159880449259385522191008031127, −7.63146003309148995139886174725, −6.55763463779863446832071002044, −5.96529762259782035141320445750, −5.21046652584928006446679121181, −4.84798696060252595465439519704, −3.53201505141896216789040613131, −2.78550979738186241033930992673, −2.28899131531695568850735053270, −1.05282040518462040320760048577,
1.05282040518462040320760048577, 2.28899131531695568850735053270, 2.78550979738186241033930992673, 3.53201505141896216789040613131, 4.84798696060252595465439519704, 5.21046652584928006446679121181, 5.96529762259782035141320445750, 6.55763463779863446832071002044, 7.63146003309148995139886174725, 8.159880449259385522191008031127
