| L(s) = 1 | + 0.806·3-s − 4.15·7-s − 2.35·9-s − 1.19·11-s + 5.35·13-s − 3.76·17-s − 6.15·19-s − 3.35·21-s + 0.806·23-s − 4.31·27-s + 29-s − 4.54·31-s − 0.962·33-s − 2.15·37-s + 4.31·39-s + 1.03·41-s + 6.41·43-s + 6.73·47-s + 10.2·49-s − 3.03·51-s + 12.8·53-s − 4.96·57-s − 6.57·59-s + 7.27·61-s + 9.76·63-s + 15.5·67-s + 0.649·69-s + ⋯ |
| L(s) = 1 | + 0.465·3-s − 1.57·7-s − 0.783·9-s − 0.359·11-s + 1.48·13-s − 0.913·17-s − 1.41·19-s − 0.731·21-s + 0.168·23-s − 0.829·27-s + 0.185·29-s − 0.816·31-s − 0.167·33-s − 0.354·37-s + 0.690·39-s + 0.162·41-s + 0.978·43-s + 0.981·47-s + 1.46·49-s − 0.425·51-s + 1.77·53-s − 0.657·57-s − 0.855·59-s + 0.931·61-s + 1.23·63-s + 1.89·67-s + 0.0782·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228510868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228510868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.806T + 3T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 23 | \( 1 - 0.806T + 23T^{2} \) |
| 31 | \( 1 + 4.54T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 6.57T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 2.57T + 71T^{2} \) |
| 73 | \( 1 - 6.80T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.420960991091668778306571104106, −7.32518373139518129932743847594, −6.59039657156940479621702990776, −6.07684867685803144114507208394, −5.45889891843528060525136875486, −4.11303815967464856815877719079, −3.67699491154203262010126638947, −2.79929170569789157885945234973, −2.13700673107348871544623810274, −0.53944618493662514556937772009,
0.53944618493662514556937772009, 2.13700673107348871544623810274, 2.79929170569789157885945234973, 3.67699491154203262010126638947, 4.11303815967464856815877719079, 5.45889891843528060525136875486, 6.07684867685803144114507208394, 6.59039657156940479621702990776, 7.32518373139518129932743847594, 8.420960991091668778306571104106
