| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 3·11-s − 6·13-s − 15-s − 21-s + 2·23-s + 25-s + 5·27-s − 6·29-s + 3·33-s + 35-s − 37-s + 6·39-s − 9·41-s − 10·43-s − 2·45-s + 47-s − 6·49-s + 53-s − 3·55-s − 12·61-s − 2·63-s − 6·65-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 1.66·13-s − 0.258·15-s − 0.218·21-s + 0.417·23-s + 1/5·25-s + 0.962·27-s − 1.11·29-s + 0.522·33-s + 0.169·35-s − 0.164·37-s + 0.960·39-s − 1.40·41-s − 1.52·43-s − 0.298·45-s + 0.145·47-s − 6/7·49-s + 0.137·53-s − 0.404·55-s − 1.53·61-s − 0.251·63-s − 0.744·65-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.06693096796852282056110210396, −9.191188851490691980399416087263, −8.157850461559496441964487907626, −7.32826710847825990750718984450, −6.32683576078001998497358732583, −5.19721952612660320591617713789, −4.94446485950534101580475152294, −3.14478969153178028838838825307, −2.04228434402964886050177056014, 0,
2.04228434402964886050177056014, 3.14478969153178028838838825307, 4.94446485950534101580475152294, 5.19721952612660320591617713789, 6.32683576078001998497358732583, 7.32826710847825990750718984450, 8.157850461559496441964487907626, 9.191188851490691980399416087263, 10.06693096796852282056110210396
