| L(s) = 1 | − 2·2-s + 2·4-s + 3·5-s − 3·9-s − 6·10-s − 6·11-s + 13-s − 4·16-s − 4·17-s + 6·18-s − 5·19-s + 6·20-s + 12·22-s + 3·23-s + 4·25-s − 2·26-s − 5·29-s + 3·31-s + 8·32-s + 8·34-s − 6·36-s − 4·37-s + 10·38-s + 6·41-s − 43-s − 12·44-s − 9·45-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.34·5-s − 9-s − 1.89·10-s − 1.80·11-s + 0.277·13-s − 16-s − 0.970·17-s + 1.41·18-s − 1.14·19-s + 1.34·20-s + 2.55·22-s + 0.625·23-s + 4/5·25-s − 0.392·26-s − 0.928·29-s + 0.538·31-s + 1.41·32-s + 1.37·34-s − 36-s − 0.657·37-s + 1.62·38-s + 0.937·41-s − 0.152·43-s − 1.80·44-s − 1.34·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−10.06879161414549558260575138607, −9.232409444628192264737671791785, −8.568760068955747319631158439108, −7.85318737682082614131812951371, −6.67298853468060181727432085852, −5.78821104763479917640596143353, −4.82450768688560173606084932648, −2.72528535271701611265854270606, −1.93185778189567456463072050241, 0,
1.93185778189567456463072050241, 2.72528535271701611265854270606, 4.82450768688560173606084932648, 5.78821104763479917640596143353, 6.67298853468060181727432085852, 7.85318737682082614131812951371, 8.568760068955747319631158439108, 9.232409444628192264737671791785, 10.06879161414549558260575138607
