| L(s) = 1 | + 2-s − 3-s + 4-s + 1.40·5-s − 6-s − 2.54·7-s + 8-s + 9-s + 1.40·10-s + 2.57·11-s − 12-s + 0.901·13-s − 2.54·14-s − 1.40·15-s + 16-s − 0.127·17-s + 18-s + 19-s + 1.40·20-s + 2.54·21-s + 2.57·22-s − 9.17·23-s − 24-s − 3.03·25-s + 0.901·26-s − 27-s − 2.54·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.627·5-s − 0.408·6-s − 0.962·7-s + 0.353·8-s + 0.333·9-s + 0.443·10-s + 0.777·11-s − 0.288·12-s + 0.250·13-s − 0.680·14-s − 0.362·15-s + 0.250·16-s − 0.0308·17-s + 0.235·18-s + 0.229·19-s + 0.313·20-s + 0.555·21-s + 0.549·22-s − 1.91·23-s − 0.204·24-s − 0.606·25-s + 0.176·26-s − 0.192·27-s − 0.481·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 + 2.54T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 0.901T + 13T^{2} \) |
| 17 | \( 1 + 0.127T + 17T^{2} \) |
| 23 | \( 1 + 9.17T + 23T^{2} \) |
| 29 | \( 1 + 9.79T + 29T^{2} \) |
| 31 | \( 1 - 6.71T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + 6.67T + 47T^{2} \) |
| 59 | \( 1 - 7.05T + 59T^{2} \) |
| 61 | \( 1 - 1.66T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 1.44T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 0.266T + 79T^{2} \) |
| 83 | \( 1 + 3.24T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.49548952117945500788767753052, −6.61170823463047281155066559975, −6.29302274442422036580940291871, −5.66560284129935759476571439017, −4.97823086101301258615637061617, −3.82900940920500410754092225427, −3.59866983345876594707340542122, −2.28166807277994895212400932906, −1.52770950815214070196058270210, 0,
1.52770950815214070196058270210, 2.28166807277994895212400932906, 3.59866983345876594707340542122, 3.82900940920500410754092225427, 4.97823086101301258615637061617, 5.66560284129935759476571439017, 6.29302274442422036580940291871, 6.61170823463047281155066559975, 7.49548952117945500788767753052
