| L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s − 2i·7-s + i·8-s − 9-s + 2i·12-s + 2i·13-s − 2·14-s + 16-s − 6i·17-s + i·18-s − 2·19-s − 4·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + 0.577i·12-s + 0.554i·13-s − 0.534·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.458·19-s − 0.872·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8816328045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8816328045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.870251897658669245079559945803, −7.64457002103717533176717913920, −7.32800026623228234537959338868, −6.51805210839746098798524053001, −5.44072145555373755610138290620, −4.43145519543671797565592333579, −3.53543313057190371931000784845, −2.33691944417049711669645687442, −1.50113692348340565457121191534, −0.32038122832832343279610805974,
1.90821831536122576289053640466, 3.44519190011522939853774918083, 4.00598051170431795522349557015, 5.09900200147427341179760128723, 5.60196014278191689248707672053, 6.45158171289118076398077490958, 7.48979139091696947488423442883, 8.360412583388371708771774049874, 8.974489503452485386829350196393, 9.613282987381695700507569608430
