| L(s) = 1 | + 2-s + 3-s + 4-s + 1.82·5-s + 6-s + 8-s + 9-s + 1.82·10-s + 1.68·11-s + 12-s + 3.77·13-s + 1.82·15-s + 16-s + 2.59·17-s + 18-s + 1.09·19-s + 1.82·20-s + 1.68·22-s + 23-s + 24-s − 1.65·25-s + 3.77·26-s + 27-s − 0.771·29-s + 1.82·30-s − 3.86·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.817·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.578·10-s + 0.506·11-s + 0.288·12-s + 1.04·13-s + 0.472·15-s + 0.250·16-s + 0.630·17-s + 0.235·18-s + 0.250·19-s + 0.408·20-s + 0.358·22-s + 0.208·23-s + 0.204·24-s − 0.331·25-s + 0.739·26-s + 0.192·27-s − 0.143·29-s + 0.333·30-s − 0.693·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.589916325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.589916325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.82T + 5T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 29 | \( 1 + 0.771T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 2.22T + 47T^{2} \) |
| 53 | \( 1 + 7.37T + 53T^{2} \) |
| 59 | \( 1 + 6.42T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 8.22T + 71T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 - 0.771T + 79T^{2} \) |
| 83 | \( 1 - 3.33T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.88168652001422038729722045092, −7.24789275830200066871279991787, −6.36116212031059271862711970592, −5.90603460714836171437570992063, −5.20463603078767512602805686309, −4.23457163350382627171843557473, −3.59988632847214976172859410917, −2.85880406239217141672989474859, −1.90830249315047160082654340972, −1.17850285262045586183958881755,
1.17850285262045586183958881755, 1.90830249315047160082654340972, 2.85880406239217141672989474859, 3.59988632847214976172859410917, 4.23457163350382627171843557473, 5.20463603078767512602805686309, 5.90603460714836171437570992063, 6.36116212031059271862711970592, 7.24789275830200066871279991787, 7.88168652001422038729722045092
