| L(s) = 1 | + 2-s + 3.21·3-s + 4-s + 5-s + 3.21·6-s − 1.49·7-s + 8-s + 7.36·9-s + 10-s − 0.492·11-s + 3.21·12-s + 13-s − 1.49·14-s + 3.21·15-s + 16-s + 0.395·17-s + 7.36·18-s − 3.24·19-s + 20-s − 4.80·21-s − 0.492·22-s + 2.92·23-s + 3.21·24-s + 25-s + 26-s + 14.0·27-s − 1.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.85·3-s + 0.5·4-s + 0.447·5-s + 1.31·6-s − 0.564·7-s + 0.353·8-s + 2.45·9-s + 0.316·10-s − 0.148·11-s + 0.929·12-s + 0.277·13-s − 0.399·14-s + 0.831·15-s + 0.250·16-s + 0.0958·17-s + 1.73·18-s − 0.744·19-s + 0.223·20-s − 1.04·21-s − 0.104·22-s + 0.609·23-s + 0.657·24-s + 0.200·25-s + 0.196·26-s + 2.70·27-s − 0.282·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.510599685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.510599685\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 3.21T + 3T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 + 0.492T + 11T^{2} \) |
| 17 | \( 1 - 0.395T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 37 | \( 1 - 0.841T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 - 4.31T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 - 0.0406T + 53T^{2} \) |
| 59 | \( 1 - 8.14T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 + 5.53T + 67T^{2} \) |
| 71 | \( 1 + 0.596T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 7.78T + 83T^{2} \) |
| 89 | \( 1 + 1.88T + 89T^{2} \) |
| 97 | \( 1 + 3.74T + 97T^{2} \) |
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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
−8.407233426478900676898428901828, −7.85980044358011606530085385309, −6.81516457673281238982198314249, −6.54042125909516529991852967315, −5.30513523083342424179086606483, −4.43013034973067084966326623770, −3.66063799666739482645074723341, −2.96322774566162497460846932744, −2.36096078908238087483957907825, −1.38162511983388643774002410257,
1.38162511983388643774002410257, 2.36096078908238087483957907825, 2.96322774566162497460846932744, 3.66063799666739482645074723341, 4.43013034973067084966326623770, 5.30513523083342424179086606483, 6.54042125909516529991852967315, 6.81516457673281238982198314249, 7.85980044358011606530085385309, 8.407233426478900676898428901828
