| L(s) = 1 | + 4·7-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s − 4·23-s − 2·29-s − 8·31-s − 6·37-s − 6·41-s + 8·43-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s − 12·63-s − 8·67-s + 6·73-s + 16·77-s + 9·81-s + 16·83-s − 6·89-s + 8·91-s + 14·97-s − 12·99-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + 1.75·83-s − 0.635·89-s + 0.838·91-s + 1.42·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327698878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327698878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.16106454520843121806724596230, −11.45666669128344280655084086532, −10.83635878381861701657899519690, −9.267253735733507250496452791400, −8.516348691113562212594470236153, −7.50932176073420730404294525891, −6.11165748014924370206663675456, −5.01978566915858300867155671122, −3.66795927025167344126272800405, −1.73185811467864323225134521099,
1.73185811467864323225134521099, 3.66795927025167344126272800405, 5.01978566915858300867155671122, 6.11165748014924370206663675456, 7.50932176073420730404294525891, 8.516348691113562212594470236153, 9.267253735733507250496452791400, 10.83635878381861701657899519690, 11.45666669128344280655084086532, 12.16106454520843121806724596230
