| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·11-s − 3·13-s + 14-s + 16-s − 4·17-s + 2·22-s − 4·23-s − 5·25-s − 3·26-s + 28-s − 3·31-s + 32-s − 4·34-s − 5·37-s − 4·41-s − 9·43-s + 2·44-s − 4·46-s − 10·47-s − 6·49-s − 5·50-s − 3·52-s + 4·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.426·22-s − 0.834·23-s − 25-s − 0.588·26-s + 0.188·28-s − 0.538·31-s + 0.176·32-s − 0.685·34-s − 0.821·37-s − 0.624·41-s − 1.37·43-s + 0.301·44-s − 0.589·46-s − 1.45·47-s − 6/7·49-s − 0.707·50-s − 0.416·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.50727163341642116560481426828, −6.85545217444089561847219939248, −6.27075428503338386215963543011, −5.38621693677820962887664692170, −4.80065633486119479778731610428, −4.03962720202051060719646779227, −3.37023256108413759660766045528, −2.23651451653199078821675184262, −1.66885606211393383518733282723, 0,
1.66885606211393383518733282723, 2.23651451653199078821675184262, 3.37023256108413759660766045528, 4.03962720202051060719646779227, 4.80065633486119479778731610428, 5.38621693677820962887664692170, 6.27075428503338386215963543011, 6.85545217444089561847219939248, 7.50727163341642116560481426828
