| L(s) = 1 | + 0.795·3-s + 3.83·5-s − 0.297·7-s − 2.36·9-s + 2.28·11-s − 6.87·13-s + 3.05·15-s + 2.62·17-s + 2.89·19-s − 0.236·21-s − 3.98·23-s + 9.73·25-s − 4.26·27-s + 3.29·29-s + 3.05·31-s + 1.81·33-s − 1.14·35-s + 9.12·37-s − 5.47·39-s − 1.23·41-s + 5.27·43-s − 9.08·45-s + 13.5·47-s − 6.91·49-s + 2.08·51-s − 0.167·53-s + 8.76·55-s + ⋯ |
| L(s) = 1 | + 0.459·3-s + 1.71·5-s − 0.112·7-s − 0.789·9-s + 0.688·11-s − 1.90·13-s + 0.788·15-s + 0.636·17-s + 0.664·19-s − 0.0516·21-s − 0.831·23-s + 1.94·25-s − 0.821·27-s + 0.612·29-s + 0.549·31-s + 0.316·33-s − 0.193·35-s + 1.50·37-s − 0.875·39-s − 0.193·41-s + 0.804·43-s − 1.35·45-s + 1.98·47-s − 0.987·49-s + 0.292·51-s − 0.0229·53-s + 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.223553203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.223553203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.795T + 3T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 + 0.297T + 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 + 6.87T + 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + 0.167T + 53T^{2} \) |
| 59 | \( 1 + 0.470T + 59T^{2} \) |
| 61 | \( 1 + 4.07T + 61T^{2} \) |
| 67 | \( 1 - 9.63T + 67T^{2} \) |
| 71 | \( 1 + 2.27T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76999300395565867506486790668, −7.19775022343683247601476660495, −6.22573155158860522950886575276, −5.86849389045054616403482752629, −5.14254894841842224556674204892, −4.41481393838097559267040230379, −3.20419345352194078941592252997, −2.54466287115801504441081352153, −2.04109839593281574432294812151, −0.864352778529562244250185853504,
0.864352778529562244250185853504, 2.04109839593281574432294812151, 2.54466287115801504441081352153, 3.20419345352194078941592252997, 4.41481393838097559267040230379, 5.14254894841842224556674204892, 5.86849389045054616403482752629, 6.22573155158860522950886575276, 7.19775022343683247601476660495, 7.76999300395565867506486790668
