| L(s) = 1 | − 2-s − 4-s + 3·7-s + 3·8-s − 11-s − 2·13-s − 3·14-s − 16-s − 3·17-s − 19-s + 22-s − 23-s + 2·26-s − 3·28-s − 6·29-s + 4·31-s − 5·32-s + 3·34-s − 37-s + 38-s + 5·41-s − 4·43-s + 44-s + 46-s − 3·47-s + 2·49-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.13·7-s + 1.06·8-s − 0.301·11-s − 0.554·13-s − 0.801·14-s − 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.213·22-s − 0.208·23-s + 0.392·26-s − 0.566·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.514·34-s − 0.164·37-s + 0.162·38-s + 0.780·41-s − 0.609·43-s + 0.150·44-s + 0.147·46-s − 0.437·47-s + 2/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.503195786802655471275015261860, −7.925108494003391656542258459331, −7.37326409570226547636119611448, −6.31018569651781389923413435679, −5.13569250388043882655281554123, −4.72076358349491808912876066304, −3.80888706948583729809072577283, −2.35667797071707152789075717627, −1.42734846445239921757802359874, 0,
1.42734846445239921757802359874, 2.35667797071707152789075717627, 3.80888706948583729809072577283, 4.72076358349491808912876066304, 5.13569250388043882655281554123, 6.31018569651781389923413435679, 7.37326409570226547636119611448, 7.925108494003391656542258459331, 8.503195786802655471275015261860
