L(s) = 1 | − 7.57·3-s + 5.40·5-s + 22.1·7-s + 30.3·9-s + 76.8·13-s − 40.9·15-s + 59.2·17-s − 95.2·19-s − 167.·21-s − 142.·23-s − 95.7·25-s − 25.6·27-s − 20.4·29-s − 213.·31-s + 119.·35-s − 145.·37-s − 582.·39-s + 82.8·41-s + 151.·43-s + 164.·45-s − 90.3·47-s + 147.·49-s − 449.·51-s − 234.·53-s + 721.·57-s − 302.·59-s − 149.·61-s + ⋯ |
L(s) = 1 | − 1.45·3-s + 0.483·5-s + 1.19·7-s + 1.12·9-s + 1.63·13-s − 0.705·15-s + 0.845·17-s − 1.15·19-s − 1.74·21-s − 1.29·23-s − 0.766·25-s − 0.183·27-s − 0.130·29-s − 1.23·31-s + 0.578·35-s − 0.646·37-s − 2.38·39-s + 0.315·41-s + 0.536·43-s + 0.544·45-s − 0.280·47-s + 0.430·49-s − 1.23·51-s − 0.608·53-s + 1.67·57-s − 0.667·59-s − 0.313·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 7.57T + 27T^{2} \) |
| 5 | \( 1 - 5.40T + 125T^{2} \) |
| 7 | \( 1 - 22.1T + 343T^{2} \) |
| 13 | \( 1 - 76.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 145.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 82.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 90.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 302.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 898.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 137.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 304.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 764.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 582.T + 9.12e5T^{2} \) |
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
−8.330675556093216504647089880377, −7.70988687114217263747472618166, −6.53134211722120207777453783178, −5.91277781717168167058725334013, −5.46791064519424492749877204619, −4.50115493962109934537005616677, −3.69208106538011908116223916334, −1.93204854835355817059576715164, −1.27636711695212601717034269464, 0,
1.27636711695212601717034269464, 1.93204854835355817059576715164, 3.69208106538011908116223916334, 4.50115493962109934537005616677, 5.46791064519424492749877204619, 5.91277781717168167058725334013, 6.53134211722120207777453783178, 7.70988687114217263747472618166, 8.330675556093216504647089880377