| L(s) = 1 | − 2·5-s − 1.41i·11-s + 7.07i·17-s − 5.65i·19-s + 1.41i·23-s − 25-s + 6·29-s − 2·31-s − 8.48i·37-s + 4.24i·41-s + (−5 + 4.24i)43-s − 1.41i·47-s + 7·49-s + 1.41i·53-s + 2.82i·55-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.426i·11-s + 1.71i·17-s − 1.29i·19-s + 0.294i·23-s − 0.200·25-s + 1.11·29-s − 0.359·31-s − 1.39i·37-s + 0.662i·41-s + (−0.762 + 0.646i)43-s − 0.206i·47-s + 49-s + 0.194i·53-s + 0.381i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.329700303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.329700303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 43 | \( 1 + (5 - 4.24i)T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 19 | \( 1 + 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 4.24iT - 59T^{2} \) |
| 61 | \( 1 + 2.82iT - 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.504086131249739847584053917668, −8.045725228202635196573269030095, −7.21929179656618413371794228214, −6.45125772872042624964667559259, −5.66582579397380973429946905057, −4.67138778513838048227146681925, −3.92525942810805146726065214759, −3.19822534735738129371931262492, −2.02435442355053038638777075840, −0.64096002281387407379303769302,
0.74553206109740259248772397539, 2.16179179194181853578539257065, 3.20856385959482307972463412995, 4.00689350101133140454196853445, 4.82887417262458704273129734312, 5.56430929533400338359064093723, 6.71384251073176095385230425537, 7.18882980138774063014594526600, 8.078771650454477985715992289794, 8.486845686518032699732768737330
