L(s) = 1 | + 2.69·2-s − 9·3-s − 24.7·4-s − 80.3·5-s − 24.2·6-s − 162.·7-s − 153.·8-s + 81·9-s − 216.·10-s − 426.·11-s + 222.·12-s − 106.·13-s − 437.·14-s + 723.·15-s + 377.·16-s − 864.·17-s + 218.·18-s − 676.·19-s + 1.98e3·20-s + 1.45e3·21-s − 1.15e3·22-s + 783.·23-s + 1.37e3·24-s + 3.33e3·25-s − 286.·26-s − 729·27-s + 4.00e3·28-s + ⋯ |
L(s) = 1 | + 0.477·2-s − 0.577·3-s − 0.772·4-s − 1.43·5-s − 0.275·6-s − 1.25·7-s − 0.845·8-s + 0.333·9-s − 0.685·10-s − 1.06·11-s + 0.445·12-s − 0.174·13-s − 0.596·14-s + 0.829·15-s + 0.368·16-s − 0.725·17-s + 0.159·18-s − 0.429·19-s + 1.10·20-s + 0.721·21-s − 0.507·22-s + 0.308·23-s + 0.488·24-s + 1.06·25-s − 0.0832·26-s − 0.192·27-s + 0.965·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.07901174122\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07901174122\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
good | 2 | \( 1 - 2.69T + 32T^{2} \) |
| 5 | \( 1 + 80.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 162.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 426.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 106.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 864.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 676.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 783.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 568.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.90e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.90e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.92e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.81e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.74e5T + 8.58e9T^{2} \) |
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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
−12.00217969574257985846035930941, −10.96281745651804983192577633747, −9.866729976217223704794199970350, −8.725238160242335239415238746130, −7.59932051067398613805544350884, −6.42262259457811725117705574659, −5.13717959438700368534521021678, −4.11106603543875836557620626645, −3.10935645830598113267951476901, −0.16061467988902900320233417912,
0.16061467988902900320233417912, 3.10935645830598113267951476901, 4.11106603543875836557620626645, 5.13717959438700368534521021678, 6.42262259457811725117705574659, 7.59932051067398613805544350884, 8.725238160242335239415238746130, 9.866729976217223704794199970350, 10.96281745651804983192577633747, 12.00217969574257985846035930941