| L(s) = 1 | + 6.42·3-s − 5·5-s − 8.32·7-s + 14.3·9-s − 1.59·11-s + 10.1·13-s − 32.1·15-s − 43.6·17-s + 48.9·19-s − 53.5·21-s + 23·23-s + 25·25-s − 81.4·27-s + 200.·29-s − 34.6·31-s − 10.2·33-s + 41.6·35-s + 92.5·37-s + 65.0·39-s + 253.·41-s − 446.·43-s − 71.6·45-s − 315.·47-s − 273.·49-s − 280.·51-s − 515.·53-s + 7.98·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 0.447·5-s − 0.449·7-s + 0.530·9-s − 0.0437·11-s + 0.215·13-s − 0.553·15-s − 0.623·17-s + 0.591·19-s − 0.555·21-s + 0.208·23-s + 0.200·25-s − 0.580·27-s + 1.28·29-s − 0.201·31-s − 0.0541·33-s + 0.200·35-s + 0.411·37-s + 0.267·39-s + 0.965·41-s − 1.58·43-s − 0.237·45-s − 0.977·47-s − 0.798·49-s − 0.771·51-s − 1.33·53-s + 0.0195·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 6.42T + 27T^{2} \) |
| 7 | \( 1 + 8.32T + 343T^{2} \) |
| 11 | \( 1 + 1.59T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 34.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 92.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 446.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 315.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 515.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 98.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 309.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 700.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 531.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 531.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 228.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 618.T + 9.12e5T^{2} \) |
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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
−8.317184876080310172360779670000, −8.036600019034601893050303719223, −7.00817635862321519039329574492, −6.30954547832114800284114442900, −5.08533584894680114971491620537, −4.14031055222023742164827108499, −3.24868136420704702967905202553, −2.68760032615726265885994976543, −1.45133861242682296717495512872, 0,
1.45133861242682296717495512872, 2.68760032615726265885994976543, 3.24868136420704702967905202553, 4.14031055222023742164827108499, 5.08533584894680114971491620537, 6.30954547832114800284114442900, 7.00817635862321519039329574492, 8.036600019034601893050303719223, 8.317184876080310172360779670000
