L(s) = 1 | + 1.01i·2-s − i·3-s + 0.969·4-s + 1.37i·5-s + 1.01·6-s + 3.01i·8-s − 9-s − 1.39·10-s + (1.32 − 3.03i)11-s − 0.969i·12-s + 0.625·13-s + 1.37·15-s − 1.11·16-s + 1.13·17-s − 1.01i·18-s − 3.04·19-s + ⋯ |
L(s) = 1 | + 0.717i·2-s − 0.577i·3-s + 0.484·4-s + 0.612i·5-s + 0.414·6-s + 1.06i·8-s − 0.333·9-s − 0.439·10-s + (0.400 − 0.916i)11-s − 0.279i·12-s + 0.173·13-s + 0.353·15-s − 0.279·16-s + 0.275·17-s − 0.239i·18-s − 0.698·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.152221003\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.152221003\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-1.32 + 3.03i)T \) |
good | 2 | \( 1 - 1.01iT - 2T^{2} \) |
| 5 | \( 1 - 1.37iT - 5T^{2} \) |
| 13 | \( 1 - 0.625T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 - 7.31T + 23T^{2} \) |
| 29 | \( 1 - 5.55iT - 29T^{2} \) |
| 31 | \( 1 + 2.26iT - 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 - 7.17T + 41T^{2} \) |
| 43 | \( 1 - 3.72iT - 43T^{2} \) |
| 47 | \( 1 - 3.00iT - 47T^{2} \) |
| 53 | \( 1 - 0.473T + 53T^{2} \) |
| 59 | \( 1 - 3.07iT - 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 4.78T + 67T^{2} \) |
| 71 | \( 1 - 3.67T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 2.26iT - 79T^{2} \) |
| 83 | \( 1 - 8.93T + 83T^{2} \) |
| 89 | \( 1 - 7.47iT - 89T^{2} \) |
| 97 | \( 1 + 14.5iT - 97T^{2} \) |
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
−9.185616517480499081449431703258, −8.581466347774663148428985324449, −7.69264562710179104589452697496, −7.08344563303184628090553722671, −6.34836083772501616202575138992, −5.86920649889604714955715628786, −4.77574558289487827350547886838, −3.29488031771630139809876098348, −2.59856380089738948838591110379, −1.21880785367106360805252743305,
0.977345611486620476080783919755, 2.16452372991148602417267157107, 3.17664094083633739641275818985, 4.19450373975035342235990789817, 4.85044792352595167851806449140, 5.99792258788857240473283669546, 6.85338432745128102404084685009, 7.68336533417641521002193843415, 8.807482137449028841893984288055, 9.369681409486007882187351899144