L(s) = 1 | − 2-s + 4-s − 8-s + 16-s + (1 − i)17-s + (1 + i)19-s + (−1 + i)23-s − 32-s + (−1 + i)34-s + (−1 − i)38-s + (1 − i)46-s + (1 − i)47-s − i·49-s + 2i·53-s + (1 − i)61-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 8-s + 16-s + (1 − i)17-s + (1 + i)19-s + (−1 + i)23-s − 32-s + (−1 + i)34-s + (−1 − i)38-s + (1 − i)46-s + (1 − i)47-s − i·49-s + 2i·53-s + (1 − i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8473083147\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8473083147\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-1 + i)T - iT^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−8.763907658950581860575461707022, −7.896773680884115952924095968678, −7.54005616866590754321907977609, −6.75788248891037997492784621781, −5.73867384980883470554502285022, −5.33240190214868689701780887518, −3.85240628417037585804803315095, −3.12711639803258165308250889241, −2.05766774731570020841204835253, −1.00039767115762042892346774083,
0.901534009153222065044194060008, 2.05762052439398049788431617748, 2.99415304506268720952436637574, 3.90029188327220822394473588323, 5.08251563771190697598871274012, 5.96427359175341310717797828612, 6.55877104087657494037607875114, 7.49948590138733128515487477591, 7.949276505332593696836538162918, 8.747506268940613382712887968863