| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 11-s + 2·13-s − 2·14-s + 16-s + 6·17-s + 2·19-s − 22-s − 5·25-s − 2·26-s + 2·28-s + 6·29-s − 4·31-s − 32-s − 6·34-s + 2·37-s − 2·38-s − 6·41-s − 10·43-s + 44-s − 12·47-s − 3·49-s + 5·50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.301·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.213·22-s − 25-s − 0.392·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.324·38-s − 0.937·41-s − 1.52·43-s + 0.150·44-s − 1.75·47-s − 3/7·49-s + 0.707·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9743637297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9743637297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.11588862961870755905569273447, −11.51202033453053796327436038455, −10.41612451305249029115472293320, −9.554119877489374041629989668544, −8.369833222628214874368506793977, −7.67970621378441729070222558677, −6.37260175111975523580528956634, −5.12309401721916407282462954040, −3.42781315245391160473547778990, −1.51420742212859032442326804401,
1.51420742212859032442326804401, 3.42781315245391160473547778990, 5.12309401721916407282462954040, 6.37260175111975523580528956634, 7.67970621378441729070222558677, 8.369833222628214874368506793977, 9.554119877489374041629989668544, 10.41612451305249029115472293320, 11.51202033453053796327436038455, 12.11588862961870755905569273447
