| L(s) = 1 | − 3·3-s + 3·7-s + 6·9-s + 5·11-s − 2·13-s − 4·17-s − 4·19-s − 9·21-s − 6·23-s − 9·27-s + 6·29-s − 4·31-s − 15·33-s + 37-s + 6·39-s − 9·41-s − 10·43-s + 11·47-s + 2·49-s + 12·51-s + 11·53-s + 12·57-s − 8·59-s − 8·61-s + 18·63-s + 8·67-s + 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.13·7-s + 2·9-s + 1.50·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s − 1.96·21-s − 1.25·23-s − 1.73·27-s + 1.11·29-s − 0.718·31-s − 2.61·33-s + 0.164·37-s + 0.960·39-s − 1.40·41-s − 1.52·43-s + 1.60·47-s + 2/7·49-s + 1.68·51-s + 1.51·53-s + 1.58·57-s − 1.04·59-s − 1.02·61-s + 2.26·63-s + 0.977·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.154290748669424388477852300389, −7.07577695780277543069908710611, −6.60981396730997064078221589991, −5.96334860425801266051119721220, −5.09888884088066200767920008704, −4.48295490912051991181364428829, −3.93322311807360794055015553666, −2.09146483456977231198455242250, −1.30728896437273467280345954384, 0,
1.30728896437273467280345954384, 2.09146483456977231198455242250, 3.93322311807360794055015553666, 4.48295490912051991181364428829, 5.09888884088066200767920008704, 5.96334860425801266051119721220, 6.60981396730997064078221589991, 7.07577695780277543069908710611, 8.154290748669424388477852300389
