| L(s) = 1 | + i·5-s + 3.60i·7-s − i·11-s + 4·13-s + (3.60 + 2i)17-s + 7·19-s − 25-s − 9i·29-s − 7.21i·31-s − 3.60·35-s + 3.60i·37-s + 5i·41-s − 2·43-s + 10.8·47-s − 5.99·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.36i·7-s − 0.301i·11-s + 1.10·13-s + (0.874 + 0.485i)17-s + 1.60·19-s − 0.200·25-s − 1.67i·29-s − 1.29i·31-s − 0.609·35-s + 0.592i·37-s + 0.780i·41-s − 0.304·43-s + 1.57·47-s − 0.857·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.104555932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.104555932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 17 | \( 1 + (-3.60 - 2i)T \) |
| good | 7 | \( 1 - 3.60iT - 7T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 7.21iT - 31T^{2} \) |
| 37 | \( 1 - 3.60iT - 37T^{2} \) |
| 41 | \( 1 - 5iT - 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 3.60T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 14.4iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8iT - 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 7.21iT - 97T^{2} \) |
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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
−8.872164811111450712589810977398, −8.015493780707871642207263556581, −7.56835568313003716172690823205, −6.18309673051646118861449868317, −6.00710067362678041573220941491, −5.20413713847180051637261087870, −3.98747243287054528475596738222, −3.15922733100283027694971670124, −2.38003464028532640244395000210, −1.13005992706493845360334356819,
0.830642958132216915661956436200, 1.52511870327712413696911063719, 3.26147689689149647557610023864, 3.67229092852187058632149825292, 4.79764974446538236953512707415, 5.36002050771621497614467552163, 6.39168470530361993539732839949, 7.34781734655947424962552129871, 7.56492986955858816190050947704, 8.654994462634191328133143707361
