| L(s) = 1 | − 1.31·2-s + 3-s − 0.280·4-s − 1.11·5-s − 1.31·6-s + 2.99·8-s + 9-s + 1.46·10-s + 2.92·11-s − 0.280·12-s + 0.149·13-s − 1.11·15-s − 3.36·16-s − 0.809·17-s − 1.31·18-s − 7.67·19-s + 0.313·20-s − 3.83·22-s − 23-s + 2.99·24-s − 3.74·25-s − 0.196·26-s + 27-s + 9.97·29-s + 1.46·30-s − 1.00·31-s − 1.57·32-s + ⋯ |
| L(s) = 1 | − 0.927·2-s + 0.577·3-s − 0.140·4-s − 0.500·5-s − 0.535·6-s + 1.05·8-s + 0.333·9-s + 0.463·10-s + 0.881·11-s − 0.0808·12-s + 0.0415·13-s − 0.288·15-s − 0.840·16-s − 0.196·17-s − 0.309·18-s − 1.76·19-s + 0.0700·20-s − 0.817·22-s − 0.208·23-s + 0.610·24-s − 0.749·25-s − 0.0385·26-s + 0.192·27-s + 1.85·29-s + 0.267·30-s − 0.180·31-s − 0.278·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 - 0.149T + 13T^{2} \) |
| 17 | \( 1 + 0.809T + 17T^{2} \) |
| 19 | \( 1 + 7.67T + 19T^{2} \) |
| 29 | \( 1 - 9.97T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 - 4.97T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.722T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.38T + 53T^{2} \) |
| 59 | \( 1 - 6.45T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 3.14T + 97T^{2} \) |
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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
−8.321719000412815467678961081383, −7.926601779194158380266556400985, −6.81711862479341932841835228298, −6.40838900224629623352510919004, −4.93621624985086463534019591649, −4.25035573330722416586333746104, −3.60014920199487992867846640822, −2.29747751688301724102113944949, −1.33781953647340100063610603109, 0,
1.33781953647340100063610603109, 2.29747751688301724102113944949, 3.60014920199487992867846640822, 4.25035573330722416586333746104, 4.93621624985086463534019591649, 6.40838900224629623352510919004, 6.81711862479341932841835228298, 7.926601779194158380266556400985, 8.321719000412815467678961081383
