L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·11-s + 13-s − 14-s + 16-s − 19-s − 3·22-s + 6·23-s + 26-s − 28-s − 6·29-s − 4·31-s + 32-s − 2·37-s − 38-s − 3·41-s + 43-s − 3·44-s + 6·46-s + 9·47-s + 49-s + 52-s − 3·53-s − 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.229·19-s − 0.639·22-s + 1.25·23-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.328·37-s − 0.162·38-s − 0.468·41-s + 0.152·43-s − 0.452·44-s + 0.884·46-s + 1.31·47-s + 1/7·49-s + 0.138·52-s − 0.412·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−7.33525161574670180844916310001, −6.62142311322130506307084822210, −5.84926691606024658291065787758, −5.33086774531301444160004851915, −4.62539689715339366614752787335, −3.78645006818420864823702282016, −3.10859672921246680827878349694, −2.39481983228610393809744814666, −1.39030009645632472393134527490, 0,
1.39030009645632472393134527490, 2.39481983228610393809744814666, 3.10859672921246680827878349694, 3.78645006818420864823702282016, 4.62539689715339366614752787335, 5.33086774531301444160004851915, 5.84926691606024658291065787758, 6.62142311322130506307084822210, 7.33525161574670180844916310001