| L(s) = 1 | + 2·3-s + 9-s + 2·11-s − 2·17-s + 2·19-s − 2·23-s − 4·27-s + 6·29-s − 2·31-s + 4·33-s + 6·37-s − 2·41-s + 6·43-s − 8·47-s − 7·49-s − 4·51-s − 2·53-s + 4·57-s + 6·59-s + 14·61-s − 4·69-s − 10·71-s − 2·73-s − 4·79-s − 11·81-s − 12·83-s + 12·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.603·11-s − 0.485·17-s + 0.458·19-s − 0.417·23-s − 0.769·27-s + 1.11·29-s − 0.359·31-s + 0.696·33-s + 0.986·37-s − 0.312·41-s + 0.914·43-s − 1.16·47-s − 49-s − 0.560·51-s − 0.274·53-s + 0.529·57-s + 0.781·59-s + 1.79·61-s − 0.481·69-s − 1.18·71-s − 0.234·73-s − 0.450·79-s − 1.22·81-s − 1.31·83-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.13177399228278, −12.74030027181053, −12.01690744188148, −11.59960821007113, −11.33004186467974, −10.64637967124803, −10.11889384586405, −9.558556804467203, −9.418188924767174, −8.753862041272362, −8.332302658318012, −8.094047523627106, −7.443518663425436, −6.956265314043923, −6.517520763202537, −5.880549596276839, −5.473399127985320, −4.590539973674457, −4.380054882653137, −3.628169479378671, −3.280053909353158, −2.662899956384502, −2.211399416826219, −1.567494902274609, −0.9246491375085745, 0,
0.9246491375085745, 1.567494902274609, 2.211399416826219, 2.662899956384502, 3.280053909353158, 3.628169479378671, 4.380054882653137, 4.590539973674457, 5.473399127985320, 5.880549596276839, 6.517520763202537, 6.956265314043923, 7.443518663425436, 8.094047523627106, 8.332302658318012, 8.753862041272362, 9.418188924767174, 9.558556804467203, 10.11889384586405, 10.64637967124803, 11.33004186467974, 11.59960821007113, 12.01690744188148, 12.74030027181053, 13.13177399228278
