| L(s) = 1 | + 2-s + 2.90·3-s + 4-s + 1.49·5-s + 2.90·6-s + 8-s + 5.45·9-s + 1.49·10-s + 1.65·11-s + 2.90·12-s − 4.81·13-s + 4.34·15-s + 16-s − 0.925·17-s + 5.45·18-s + 4.40·19-s + 1.49·20-s + 1.65·22-s − 0.691·23-s + 2.90·24-s − 2.76·25-s − 4.81·26-s + 7.15·27-s − 29-s + 4.34·30-s + 7.15·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s + 0.668·5-s + 1.18·6-s + 0.353·8-s + 1.81·9-s + 0.472·10-s + 0.498·11-s + 0.839·12-s − 1.33·13-s + 1.12·15-s + 0.250·16-s − 0.224·17-s + 1.28·18-s + 1.01·19-s + 0.334·20-s + 0.352·22-s − 0.144·23-s + 0.593·24-s − 0.553·25-s − 0.943·26-s + 1.37·27-s − 0.185·29-s + 0.793·30-s + 1.28·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.981638485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.981638485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + 0.925T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 + 0.691T + 23T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 8.91T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 7.87T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−8.875739681962330892557968095731, −7.88509749802771235187641177423, −7.41241540256686373430481079275, −6.60402112265562415181121246008, −5.62852272132215054181166810056, −4.68725723757493490708764875030, −3.93378573157589509064851509659, −2.97307187068596459852382407160, −2.41694688383883644122127003916, −1.50911126064617855427723972297,
1.50911126064617855427723972297, 2.41694688383883644122127003916, 2.97307187068596459852382407160, 3.93378573157589509064851509659, 4.68725723757493490708764875030, 5.62852272132215054181166810056, 6.60402112265562415181121246008, 7.41241540256686373430481079275, 7.88509749802771235187641177423, 8.875739681962330892557968095731
