L(s) = 1 | − 9.15·5-s − 27.4·7-s − 20.5·11-s − 32.0·13-s + 111.·17-s + 129.·19-s + 9.16·23-s − 41.1·25-s + 41.0·29-s − 187.·31-s + 251.·35-s + 114.·37-s + 282.·41-s − 89.3·43-s + 54.6·47-s + 408.·49-s + 726.·53-s + 187.·55-s + 216.·59-s − 754.·61-s + 293.·65-s + 379.·67-s + 302.·71-s + 504.·73-s + 562.·77-s + 301.·79-s − 599.·83-s + ⋯ |
L(s) = 1 | − 0.818·5-s − 1.48·7-s − 0.562·11-s − 0.683·13-s + 1.59·17-s + 1.56·19-s + 0.0830·23-s − 0.329·25-s + 0.262·29-s − 1.08·31-s + 1.21·35-s + 0.507·37-s + 1.07·41-s − 0.317·43-s + 0.169·47-s + 1.19·49-s + 1.88·53-s + 0.460·55-s + 0.477·59-s − 1.58·61-s + 0.559·65-s + 0.691·67-s + 0.504·71-s + 0.808·73-s + 0.832·77-s + 0.429·79-s − 0.792·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 9.15T + 125T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 11 | \( 1 + 20.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.16T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 89.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 54.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 754.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 599.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 765.T + 9.12e5T^{2} \) |
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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
−8.030114330952035488021400952348, −7.51920276436931709981823479390, −6.92068287342766729979839919191, −5.78206540293048254433526659614, −5.26836639765476518793701593507, −3.98099867076858959594112509507, −3.32374310164769577599957698701, −2.62364377766526440657683146098, −0.954202248102399292379190874149, 0,
0.954202248102399292379190874149, 2.62364377766526440657683146098, 3.32374310164769577599957698701, 3.98099867076858959594112509507, 5.26836639765476518793701593507, 5.78206540293048254433526659614, 6.92068287342766729979839919191, 7.51920276436931709981823479390, 8.030114330952035488021400952348