| L(s) = 1 | + 0.804·3-s − 0.337·5-s − 7-s − 2.35·9-s − 11-s + 13-s − 0.271·15-s − 1.72·17-s + 3.84·19-s − 0.804·21-s − 1.18·23-s − 4.88·25-s − 4.30·27-s + 8.19·29-s + 4.39·31-s − 0.804·33-s + 0.337·35-s − 8.55·37-s + 0.804·39-s + 3.35·41-s + 7.49·43-s + 0.794·45-s − 4.05·47-s + 49-s − 1.38·51-s − 8.73·53-s + 0.337·55-s + ⋯ |
| L(s) = 1 | + 0.464·3-s − 0.151·5-s − 0.377·7-s − 0.784·9-s − 0.301·11-s + 0.277·13-s − 0.0701·15-s − 0.418·17-s + 0.882·19-s − 0.175·21-s − 0.246·23-s − 0.977·25-s − 0.828·27-s + 1.52·29-s + 0.788·31-s − 0.140·33-s + 0.0571·35-s − 1.40·37-s + 0.128·39-s + 0.523·41-s + 1.14·43-s + 0.118·45-s − 0.591·47-s + 0.142·49-s − 0.194·51-s − 1.19·53-s + 0.0455·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.698259220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698259220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.804T + 3T^{2} \) |
| 5 | \( 1 + 0.337T + 5T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 - 3.84T + 19T^{2} \) |
| 23 | \( 1 + 1.18T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 37 | \( 1 + 8.55T + 37T^{2} \) |
| 41 | \( 1 - 3.35T + 41T^{2} \) |
| 43 | \( 1 - 7.49T + 43T^{2} \) |
| 47 | \( 1 + 4.05T + 47T^{2} \) |
| 53 | \( 1 + 8.73T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 + 5.01T + 67T^{2} \) |
| 71 | \( 1 + 0.791T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 + 9.95T + 97T^{2} \) |
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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
−7.940340554502072953640196154528, −7.23947691675076002931434009037, −6.39544148757748736910915593088, −5.83540636627998697972267381622, −5.04230975186315051211132185811, −4.21139640986237781091668732431, −3.33753819376318005737507934374, −2.82072390576655249210244625855, −1.91241804868690158282983342219, −0.61030206952964838866446437987,
0.61030206952964838866446437987, 1.91241804868690158282983342219, 2.82072390576655249210244625855, 3.33753819376318005737507934374, 4.21139640986237781091668732431, 5.04230975186315051211132185811, 5.83540636627998697972267381622, 6.39544148757748736910915593088, 7.23947691675076002931434009037, 7.940340554502072953640196154528
