| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 12-s − 4·13-s + 2·15-s + 16-s − 18-s − 6·19-s + 2·20-s − 23-s − 24-s − 25-s + 4·26-s + 27-s − 6·29-s − 2·30-s + 10·31-s − 32-s + 36-s − 6·37-s + 6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 1.10·13-s + 0.516·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.447·20-s − 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.192·27-s − 1.11·29-s − 0.365·30-s + 1.79·31-s − 0.176·32-s + 1/6·36-s − 0.986·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.75307857458158543088604731601, −7.02738940417009546271130138998, −6.35032556384238417172028370731, −5.67932025773096788829582180097, −4.74849680205857491308806811466, −3.94095715245293709767887032654, −2.76794772888261448463062802969, −2.25988769128618149359019770536, −1.47004426395622444290709075383, 0,
1.47004426395622444290709075383, 2.25988769128618149359019770536, 2.76794772888261448463062802969, 3.94095715245293709767887032654, 4.74849680205857491308806811466, 5.67932025773096788829582180097, 6.35032556384238417172028370731, 7.02738940417009546271130138998, 7.75307857458158543088604731601
