L(s) = 1 | − i·2-s − 4-s + (−1 − i)5-s + i·8-s + i·9-s + (−1 + i)10-s + 2·13-s + 16-s − i·17-s + 18-s + (1 + i)20-s + i·25-s − 2i·26-s + (−1 − i)29-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−1 − i)5-s + i·8-s + i·9-s + (−1 + i)10-s + 2·13-s + 16-s − i·17-s + 18-s + (1 + i)20-s + i·25-s − 2i·26-s + (−1 − i)29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9410712447\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9410712447\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 17 | \( 1 + iT \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - 2T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−8.548549548274563564144906219624, −8.098908181856115567616957731689, −7.41986238014984211437602090300, −6.00542313895195228226626409227, −5.26087062239580847145273039016, −4.40871171970737961750501582965, −3.94905446137667590140928639264, −2.96896215101482196034074936860, −1.77697439988707306132956663952, −0.71656054501944727655211663883,
1.15790871148051782974946018628, 3.11611928451943545657526856909, 3.85246361645720185831238690570, 4.13088693244731173451210324301, 5.69325487277978777709354685503, 6.15038956349376653971524218849, 6.84314331405163844154461594750, 7.48358014587181380502623646717, 8.274342717562745769936631149975, 8.790390094214946818624091041978