L(s) = 1 | + i·3-s − 5-s + 8·9-s + 17i·11-s − 24·13-s − i·15-s − 17-s − 7i·19-s − 7i·23-s − 24·25-s + 17i·27-s + 24·29-s − 41i·31-s − 17·33-s − 49·37-s + ⋯ |
L(s) = 1 | + 0.333i·3-s − 0.200·5-s + 0.888·9-s + 1.54i·11-s − 1.84·13-s − 0.0666i·15-s − 0.0588·17-s − 0.368i·19-s − 0.304i·23-s − 0.959·25-s + 0.629i·27-s + 0.827·29-s − 1.32i·31-s − 0.515·33-s − 1.32·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1427410258\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1427410258\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT - 9T^{2} \) |
| 5 | \( 1 + T + 25T^{2} \) |
| 11 | \( 1 - 17iT - 121T^{2} \) |
| 13 | \( 1 + 24T + 169T^{2} \) |
| 17 | \( 1 + T + 289T^{2} \) |
| 19 | \( 1 + 7iT - 361T^{2} \) |
| 23 | \( 1 + 7iT - 529T^{2} \) |
| 29 | \( 1 - 24T + 841T^{2} \) |
| 31 | \( 1 + 41iT - 961T^{2} \) |
| 37 | \( 1 + 49T + 1.36e3T^{2} \) |
| 41 | \( 1 - 48T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 55iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25T + 2.80e3T^{2} \) |
| 59 | \( 1 - 17iT - 3.48e3T^{2} \) |
| 61 | \( 1 - T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 96iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 95T + 5.32e3T^{2} \) |
| 79 | \( 1 - 41iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 72iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 95T + 7.92e3T^{2} \) |
| 97 | \( 1 + 144T + 9.40e3T^{2} \) |
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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
−9.208845359957239238337697286015, −7.973158524482111106585256454299, −7.26532360147701610936480437099, −6.83378794036345885091336655283, −5.44964170641966762550519063799, −4.55318899343814814947202738878, −4.18926273854097998308499131918, −2.67428628889902789518317161807, −1.79748825101306848895504491385, −0.03867552626891854107382691626,
1.26876705039339875343631158995, 2.52668993365403805641694438733, 3.53569745841225698389206515271, 4.56957747946361817481303712848, 5.44252710686740174706948520646, 6.39024359585340319699812337716, 7.24493519917293213811890555274, 7.82994512642282414238231101193, 8.684249048673170416666209227502, 9.585671589860434202168201371447