| L(s) = 1 | + i·3-s + (−1.48 − 1.67i)5-s − i·7-s − 9-s − 2·11-s + 1.35i·13-s + (1.67 − 1.48i)15-s + 3.35i·17-s + 5.35·19-s + 21-s − 4.96i·23-s + (−0.612 + 4.96i)25-s − i·27-s − 7.92·29-s − 4.57·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.662 − 0.749i)5-s − 0.377i·7-s − 0.333·9-s − 0.603·11-s + 0.374i·13-s + (0.432 − 0.382i)15-s + 0.812i·17-s + 1.22·19-s + 0.218·21-s − 1.03i·23-s + (−0.122 + 0.992i)25-s − 0.192i·27-s − 1.47·29-s − 0.821·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6015597493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6015597493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 1.35iT - 13T^{2} \) |
| 17 | \( 1 - 3.35iT - 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 + 4.96iT - 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 0.775iT - 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 - 12.6iT - 43T^{2} \) |
| 47 | \( 1 - 9.92iT - 47T^{2} \) |
| 53 | \( 1 - 8.57iT - 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 9.92iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 9.35iT - 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 - 18.4iT - 97T^{2} \) |
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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
−9.367888246853116659315416354685, −9.105350712553769506833067461085, −7.79766294316717871363515221120, −7.72047531327429551191753963209, −6.32140997624745520931678555135, −5.41062421564043324500928430482, −4.58201445019367074307619770349, −3.92452534324292257502031178352, −2.92114043076329080577023644199, −1.33895383511386401285699749202,
0.23563322294345529994145419479, 1.96028284945980152550128341701, 3.03039466229066130997813957209, 3.71544286381283675176396632105, 5.21619207035762950330610227702, 5.70648654268914698712469585562, 6.96310661805904065525807764614, 7.43168858831877222904416355019, 8.006008821325226433172550139599, 9.057049496970721009047241266648
