| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s + 2·13-s + 2·17-s + 2·19-s − 2·21-s − 2·23-s + 27-s − 6·29-s + 4·31-s − 2·33-s − 2·37-s + 2·39-s − 10·41-s + 8·43-s − 2·47-s − 3·49-s + 2·51-s − 6·53-s + 2·57-s + 2·59-s − 10·61-s − 2·63-s − 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.436·21-s − 0.417·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s − 0.328·37-s + 0.320·39-s − 1.56·41-s + 1.21·43-s − 0.291·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s + 0.264·57-s + 0.260·59-s − 1.28·61-s − 0.251·63-s − 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.50020185064504762214979396718, −6.66228042823227413699168639621, −6.04888719753575154909360593500, −5.31972156976567417803960425033, −4.51040170105691037881694992213, −3.54495738735222379216715871256, −3.19544037947371539977701842610, −2.26142847899965123185978683952, −1.30735333180955841834797761707, 0,
1.30735333180955841834797761707, 2.26142847899965123185978683952, 3.19544037947371539977701842610, 3.54495738735222379216715871256, 4.51040170105691037881694992213, 5.31972156976567417803960425033, 6.04888719753575154909360593500, 6.66228042823227413699168639621, 7.50020185064504762214979396718
