| L(s) = 1 | − i·3-s + i·7-s + 2·9-s − 3·11-s + 2i·13-s − 3i·17-s − 7·19-s + 21-s − 5i·27-s + 6·29-s + 4·31-s + 3i·33-s − 8i·37-s + 2·39-s − 9·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s + 0.666·9-s − 0.904·11-s + 0.554i·13-s − 0.727i·17-s − 1.60·19-s + 0.218·21-s − 0.962i·27-s + 1.11·29-s + 0.718·31-s + 0.522i·33-s − 1.31i·37-s + 0.320·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.201589651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201589651\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 5iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.439535071527301795498875791572, −7.86524564748114601378557105422, −6.83119856000446074639433272428, −6.59758426456412457724930490957, −5.42292008102708855710365113617, −4.70439236326913879374483276491, −3.78897014058030505783461895990, −2.50515243931651445378355835746, −1.90377924917026578341818335447, −0.39297841375596980079219769807,
1.26353506707511164888895936092, 2.55441391864240419216153383205, 3.49676909267743451768052399893, 4.50931814997606011834262642095, 4.84924149369112346935189782289, 6.08198688056887357232281118306, 6.65520217491701340549277577520, 7.71813709941001194391347837439, 8.234510508171584789089015834907, 9.017369554046076493311831701931
