| L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·13-s − 6·17-s + 2·19-s − 4·21-s + 23-s − 27-s + 6·29-s − 4·31-s − 8·37-s + 2·39-s + 6·41-s − 8·43-s − 12·47-s + 9·49-s + 6·51-s + 6·53-s − 2·57-s − 6·59-s − 10·61-s + 4·63-s − 8·67-s − 69-s − 6·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 0.458·19-s − 0.872·21-s + 0.208·23-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 1.31·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.264·57-s − 0.781·59-s − 1.28·61-s + 0.503·63-s − 0.977·67-s − 0.120·69-s − 0.712·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.53802073357773735384666104050, −6.96452450971908123162486622031, −6.21861642268241639495963461965, −5.30208224624501127586076082497, −4.78103848671925512221809679715, −4.32839538753549098596341982380, −3.13249560577782150747345922132, −2.05878307656407186658862024551, −1.37808731668133455407499312232, 0,
1.37808731668133455407499312232, 2.05878307656407186658862024551, 3.13249560577782150747345922132, 4.32839538753549098596341982380, 4.78103848671925512221809679715, 5.30208224624501127586076082497, 6.21861642268241639495963461965, 6.96452450971908123162486622031, 7.53802073357773735384666104050
