| L(s) = 1 | + 0.238·3-s + 5·5-s − 7·7-s − 26.9·9-s + 54.5·11-s + 52.2·13-s + 1.19·15-s + 49.8·17-s − 161.·19-s − 1.67·21-s + 119.·23-s + 25·25-s − 12.8·27-s − 49.5·29-s + 164.·31-s + 13.0·33-s − 35·35-s + 216.·37-s + 12.4·39-s − 252.·41-s − 68.7·43-s − 134.·45-s + 84.4·47-s + 49·49-s + 11.9·51-s + 204.·53-s + 272.·55-s + ⋯ |
| L(s) = 1 | + 0.0459·3-s + 0.447·5-s − 0.377·7-s − 0.997·9-s + 1.49·11-s + 1.11·13-s + 0.0205·15-s + 0.711·17-s − 1.94·19-s − 0.0173·21-s + 1.08·23-s + 0.200·25-s − 0.0917·27-s − 0.317·29-s + 0.955·31-s + 0.0687·33-s − 0.169·35-s + 0.963·37-s + 0.0511·39-s − 0.962·41-s − 0.243·43-s − 0.446·45-s + 0.261·47-s + 0.142·49-s + 0.0327·51-s + 0.530·53-s + 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.513935324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.513935324\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 0.238T + 27T^{2} \) |
| 11 | \( 1 - 54.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 161.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 68.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 84.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 204.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 552.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 178.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 415.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 931.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 223.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.70e3T + 9.12e5T^{2} \) |
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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
−8.762416810017620780850464585016, −8.167179021623992209274150345340, −6.87588762191671394240489224048, −6.27283719092285085727225705819, −5.81813978944945175482498154655, −4.60748007361281771581016931272, −3.71278632041868335831349107935, −2.91427412289996974250598380617, −1.74982907255937896347060241540, −0.73072030463426750476499246996,
0.73072030463426750476499246996, 1.74982907255937896347060241540, 2.91427412289996974250598380617, 3.71278632041868335831349107935, 4.60748007361281771581016931272, 5.81813978944945175482498154655, 6.27283719092285085727225705819, 6.87588762191671394240489224048, 8.167179021623992209274150345340, 8.762416810017620780850464585016
