| L(s) = 1 | + 2-s + 2.89·3-s + 4-s − 5-s + 2.89·6-s − 1.48·7-s + 8-s + 5.38·9-s − 10-s − 11-s + 2.89·12-s + 3.40·13-s − 1.48·14-s − 2.89·15-s + 16-s − 17-s + 5.38·18-s + 6.38·19-s − 20-s − 4.30·21-s − 22-s + 0.513·23-s + 2.89·24-s + 25-s + 3.40·26-s + 6.89·27-s − 1.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s − 0.447·5-s + 1.18·6-s − 0.561·7-s + 0.353·8-s + 1.79·9-s − 0.316·10-s − 0.301·11-s + 0.835·12-s + 0.945·13-s − 0.397·14-s − 0.747·15-s + 0.250·16-s − 0.242·17-s + 1.26·18-s + 1.46·19-s − 0.223·20-s − 0.939·21-s − 0.213·22-s + 0.107·23-s + 0.590·24-s + 0.200·25-s + 0.668·26-s + 1.32·27-s − 0.280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.564867254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.564867254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 0.513T + 23T^{2} \) |
| 29 | \( 1 - 7.40T + 29T^{2} \) |
| 31 | \( 1 + 1.40T + 31T^{2} \) |
| 37 | \( 1 - 3.79T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 + 2.89T + 53T^{2} \) |
| 59 | \( 1 - 0.0779T + 59T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 1.23T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.126527617190964440988409192929, −8.368955993719948359721257300492, −7.73400433296078388706107315131, −7.00091442280607816100923779545, −6.11782526304937756455662406191, −4.91703750109573740307997331372, −3.98943051310844011636657202624, −3.23031675534933022062286204473, −2.76527961894552980105993367249, −1.40080054501754303770996369172,
1.40080054501754303770996369172, 2.76527961894552980105993367249, 3.23031675534933022062286204473, 3.98943051310844011636657202624, 4.91703750109573740307997331372, 6.11782526304937756455662406191, 7.00091442280607816100923779545, 7.73400433296078388706107315131, 8.368955993719948359721257300492, 9.126527617190964440988409192929
