L(s) = 1 | − 1.63·2-s + 3-s + 0.674·4-s + (−1.84 + 1.25i)5-s − 1.63·6-s + (1.79 + 1.94i)7-s + 2.16·8-s + 9-s + (3.02 − 2.05i)10-s + (−3.27 + 0.520i)11-s + 0.674·12-s − 4.17i·13-s + (−2.93 − 3.18i)14-s + (−1.84 + 1.25i)15-s − 4.89·16-s − 7.58i·17-s + ⋯ |
L(s) = 1 | − 1.15·2-s + 0.577·3-s + 0.337·4-s + (−0.826 + 0.562i)5-s − 0.667·6-s + (0.677 + 0.735i)7-s + 0.766·8-s + 0.333·9-s + (0.956 − 0.650i)10-s + (−0.987 + 0.156i)11-s + 0.194·12-s − 1.15i·13-s + (−0.783 − 0.850i)14-s + (−0.477 + 0.324i)15-s − 1.22·16-s − 1.83i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8775427553\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8775427553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (1.84 - 1.25i)T \) |
| 7 | \( 1 + (-1.79 - 1.94i)T \) |
| 11 | \( 1 + (3.27 - 0.520i)T \) |
good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 13 | \( 1 + 4.17iT - 13T^{2} \) |
| 17 | \( 1 + 7.58iT - 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + 2.17iT - 23T^{2} \) |
| 29 | \( 1 - 5.39iT - 29T^{2} \) |
| 31 | \( 1 - 0.864iT - 31T^{2} \) |
| 37 | \( 1 - 2.91iT - 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 10.8iT - 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 10.6iT - 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 + 2.25iT - 73T^{2} \) |
| 79 | \( 1 - 8.24iT - 79T^{2} \) |
| 83 | \( 1 + 5.54iT - 83T^{2} \) |
| 89 | \( 1 + 1.62iT - 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−9.651893218694973671597103796209, −8.884175794605241794952124010471, −8.127303744124628319848680316342, −7.59682171782022867630780569976, −7.09953791562406574479246652249, −5.36813721209083730633529415653, −4.71129452030315524275473359710, −3.17424213907856225562083774644, −2.45718488193496475860923265721, −0.74484148823121764551990518113,
0.941731665065792186263304091779, 2.01940717722340551785256801215, 3.81121482225734877952954293890, 4.33964503616096132332878796159, 5.46408233711186117465177739464, 7.11185189219066030232924786900, 7.62844984829895818269968083108, 8.276699580387172100017203505310, 8.761560022940108533637868236391, 9.706265637881389477103236202978