| L(s) = 1 | + 2-s + 3-s + 4-s + 0.871·5-s + 6-s + 1.79·7-s + 8-s + 9-s + 0.871·10-s + 11-s + 12-s + 6.05·13-s + 1.79·14-s + 0.871·15-s + 16-s + 4.28·17-s + 18-s − 0.502·19-s + 0.871·20-s + 1.79·21-s + 22-s − 0.329·23-s + 24-s − 4.24·25-s + 6.05·26-s + 27-s + 1.79·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.389·5-s + 0.408·6-s + 0.677·7-s + 0.353·8-s + 0.333·9-s + 0.275·10-s + 0.301·11-s + 0.288·12-s + 1.67·13-s + 0.478·14-s + 0.225·15-s + 0.250·16-s + 1.03·17-s + 0.235·18-s − 0.115·19-s + 0.194·20-s + 0.390·21-s + 0.213·22-s − 0.0686·23-s + 0.204·24-s − 0.848·25-s + 1.18·26-s + 0.192·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.063181507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.063181507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 - 0.871T + 5T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 13 | \( 1 - 6.05T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 + 0.502T + 19T^{2} \) |
| 23 | \( 1 + 0.329T + 23T^{2} \) |
| 29 | \( 1 - 0.826T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 - 0.198T + 53T^{2} \) |
| 59 | \( 1 + 5.77T + 59T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 + 7.63T + 71T^{2} \) |
| 73 | \( 1 - 7.93T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 5.33T + 83T^{2} \) |
| 89 | \( 1 + 4.51T + 89T^{2} \) |
| 97 | \( 1 - 9.53T + 97T^{2} \) |
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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
−8.285991155372446622166560903016, −7.87043896815039480677559110008, −6.82972007900814041715055774495, −6.17479565279631307392091837202, −5.44584322844686584390204802668, −4.64501115389542575382755736215, −3.68392690007273434306434139564, −3.23661165563173644875372829011, −1.92683784393446946269315239326, −1.32736676760791531120739594514,
1.32736676760791531120739594514, 1.92683784393446946269315239326, 3.23661165563173644875372829011, 3.68392690007273434306434139564, 4.64501115389542575382755736215, 5.44584322844686584390204802668, 6.17479565279631307392091837202, 6.82972007900814041715055774495, 7.87043896815039480677559110008, 8.285991155372446622166560903016
