| L(s) = 1 | + 5.77·3-s + 5·5-s + 12.7·7-s + 6.31·9-s − 51.7·11-s + 33.9·13-s + 28.8·15-s − 86.4·17-s − 59.7·19-s + 73.7·21-s + 23·23-s + 25·25-s − 119.·27-s − 64.5·29-s + 157.·31-s − 298.·33-s + 63.8·35-s − 275.·37-s + 195.·39-s − 482.·41-s + 270.·43-s + 31.5·45-s + 320.·47-s − 179.·49-s − 499.·51-s − 122.·53-s − 258.·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.447·5-s + 0.689·7-s + 0.233·9-s − 1.41·11-s + 0.724·13-s + 0.496·15-s − 1.23·17-s − 0.721·19-s + 0.766·21-s + 0.208·23-s + 0.200·25-s − 0.850·27-s − 0.413·29-s + 0.910·31-s − 1.57·33-s + 0.308·35-s − 1.22·37-s + 0.804·39-s − 1.83·41-s + 0.961·43-s + 0.104·45-s + 0.995·47-s − 0.524·49-s − 1.37·51-s − 0.317·53-s − 0.633·55-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 | \( 1 - 5.77T + 27T^{2} \) |
| 7 | \( 1 - 12.7T + 343T^{2} \) |
| 11 | \( 1 + 51.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 64.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 482.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 270.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 320.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 34.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 90.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 182.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 73.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 283.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 476.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.82e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.532191546852113821629679138409, −7.976400219246335138037263665791, −7.07731556967921706899721246511, −6.08256747155476548723584700313, −5.16808344007236317121424001798, −4.33503444431509429041727807357, −3.21283793617400360688332770152, −2.39516538415409191031470514782, −1.67709716860612405566624717637, 0,
1.67709716860612405566624717637, 2.39516538415409191031470514782, 3.21283793617400360688332770152, 4.33503444431509429041727807357, 5.16808344007236317121424001798, 6.08256747155476548723584700313, 7.07731556967921706899721246511, 7.976400219246335138037263665791, 8.532191546852113821629679138409
