| L(s) = 1 | + 2·2-s + 3.83·3-s + 4·4-s + 17.7·5-s + 7.66·6-s + 8·8-s − 12.3·9-s + 35.4·10-s − 12.8·11-s + 15.3·12-s + 1.76·13-s + 67.8·15-s + 16·16-s − 35.3·17-s − 24.6·18-s + 23.6·19-s + 70.8·20-s − 25.7·22-s + 23·23-s + 30.6·24-s + 188.·25-s + 3.52·26-s − 150.·27-s + 308.·29-s + 135.·30-s + 120.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.737·3-s + 0.5·4-s + 1.58·5-s + 0.521·6-s + 0.353·8-s − 0.455·9-s + 1.11·10-s − 0.352·11-s + 0.368·12-s + 0.0375·13-s + 1.16·15-s + 0.250·16-s − 0.504·17-s − 0.322·18-s + 0.285·19-s + 0.791·20-s − 0.249·22-s + 0.208·23-s + 0.260·24-s + 1.50·25-s + 0.0265·26-s − 1.07·27-s + 1.97·29-s + 0.825·30-s + 0.699·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.660453322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.660453322\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 3.83T + 27T^{2} \) |
| 5 | \( 1 - 17.7T + 125T^{2} \) |
| 11 | \( 1 + 12.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.76T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 308.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 437.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 468.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 409.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 637.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 624.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 435.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 988.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 524.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−8.573588224306806317350195736679, −8.103533064680461215620058804231, −6.79433233339043765491548721557, −6.34176438910266933260076732180, −5.42040676504365972335607125670, −4.87137671051040712925231759977, −3.64111925812091992747728191041, −2.58031437528567326535797481957, −2.32731510775758383792327888219, −1.03391425444622521761545393662,
1.03391425444622521761545393662, 2.32731510775758383792327888219, 2.58031437528567326535797481957, 3.64111925812091992747728191041, 4.87137671051040712925231759977, 5.42040676504365972335607125670, 6.34176438910266933260076732180, 6.79433233339043765491548721557, 8.103533064680461215620058804231, 8.573588224306806317350195736679
