| L(s) = 1 | + 3-s + 7-s + 9-s + 3·11-s − 13-s − 7·17-s + 21-s − 7·23-s + 27-s − 4·29-s + 8·31-s + 3·33-s − 5·37-s − 39-s − 3·41-s − 8·43-s + 6·47-s − 6·49-s − 7·51-s − 11·53-s − 4·59-s + 61-s + 63-s + 12·67-s − 7·69-s − 9·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s + 0.218·21-s − 1.45·23-s + 0.192·27-s − 0.742·29-s + 1.43·31-s + 0.522·33-s − 0.821·37-s − 0.160·39-s − 0.468·41-s − 1.21·43-s + 0.875·47-s − 6/7·49-s − 0.980·51-s − 1.51·53-s − 0.520·59-s + 0.128·61-s + 0.125·63-s + 1.46·67-s − 0.842·69-s − 1.06·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.61744070848064466249311358956, −6.66829151979091858079197916188, −6.43068742573207513663124904351, −5.32764273551975483527464339315, −4.47201915511794813566464693222, −4.03379937936608528946586711485, −3.09993684713890123852628980438, −2.13820228431270433295973167861, −1.53595124430068697367324888802, 0,
1.53595124430068697367324888802, 2.13820228431270433295973167861, 3.09993684713890123852628980438, 4.03379937936608528946586711485, 4.47201915511794813566464693222, 5.32764273551975483527464339315, 6.43068742573207513663124904351, 6.66829151979091858079197916188, 7.61744070848064466249311358956
