| L(s) = 1 | + 3-s − 2·9-s − 11-s − 2·13-s + 3·17-s + 3·19-s − 6·23-s − 5·27-s − 2·29-s − 2·31-s − 33-s + 4·37-s − 2·39-s − 3·41-s − 4·43-s + 6·47-s − 7·49-s + 3·51-s − 10·53-s + 3·57-s + 12·59-s − 12·61-s − 67-s − 6·69-s − 10·71-s + 73-s − 16·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.301·11-s − 0.554·13-s + 0.727·17-s + 0.688·19-s − 1.25·23-s − 0.962·27-s − 0.371·29-s − 0.359·31-s − 0.174·33-s + 0.657·37-s − 0.320·39-s − 0.468·41-s − 0.609·43-s + 0.875·47-s − 49-s + 0.420·51-s − 1.37·53-s + 0.397·57-s + 1.56·59-s − 1.53·61-s − 0.122·67-s − 0.722·69-s − 1.18·71-s + 0.117·73-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 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
−8.074976683238131552132290761167, −7.84386856218404514345521237249, −6.91321376395927162267466338886, −5.87497740088439575573494044931, −5.35774889904735159947508759505, −4.31239757720528279779934093934, −3.35479325900206951249807212451, −2.68742681871509428202308350696, −1.63983092209844712290585848003, 0,
1.63983092209844712290585848003, 2.68742681871509428202308350696, 3.35479325900206951249807212451, 4.31239757720528279779934093934, 5.35774889904735159947508759505, 5.87497740088439575573494044931, 6.91321376395927162267466338886, 7.84386856218404514345521237249, 8.074976683238131552132290761167
