| L(s) = 1 | + 3-s + 9-s − 2·11-s − 2·13-s − 4·17-s + 4·19-s + 6·23-s − 5·25-s + 27-s + 2·29-s − 2·33-s + 6·37-s − 2·39-s − 8·41-s − 8·43-s − 4·47-s − 4·51-s + 6·53-s + 4·57-s − 14·61-s + 4·67-s + 6·69-s + 2·71-s + 2·73-s − 5·75-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.917·19-s + 1.25·23-s − 25-s + 0.192·27-s + 0.371·29-s − 0.348·33-s + 0.986·37-s − 0.320·39-s − 1.24·41-s − 1.21·43-s − 0.583·47-s − 0.560·51-s + 0.824·53-s + 0.529·57-s − 1.79·61-s + 0.488·67-s + 0.722·69-s + 0.237·71-s + 0.234·73-s − 0.577·75-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.41316834474548691555357375667, −6.83364278896222152523375167060, −6.04976907103984301589697009940, −5.08821231226396920453742081831, −4.71093761217536498900331735499, −3.71043880881469772089910145016, −2.96909066521507484542218671027, −2.31698807390248258603905073049, −1.34156614571833066868188465618, 0,
1.34156614571833066868188465618, 2.31698807390248258603905073049, 2.96909066521507484542218671027, 3.71043880881469772089910145016, 4.71093761217536498900331735499, 5.08821231226396920453742081831, 6.04976907103984301589697009940, 6.83364278896222152523375167060, 7.41316834474548691555357375667
