| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s − 3·7-s − 8-s + 9-s + 4·10-s + 2·11-s + 12-s + 7·13-s + 3·14-s − 4·15-s + 16-s − 18-s − 4·20-s − 3·21-s − 2·22-s − 4·23-s − 24-s + 11·25-s − 7·26-s + 27-s − 3·28-s − 4·29-s + 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.603·11-s + 0.288·12-s + 1.94·13-s + 0.801·14-s − 1.03·15-s + 1/4·16-s − 0.235·18-s − 0.894·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.204·24-s + 11/5·25-s − 1.37·26-s + 0.192·27-s − 0.566·28-s − 0.742·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.632336773486661306730920493015, −8.119209499510440553100289543051, −7.23066919257894410409371315840, −6.68461495565388836489640760645, −5.76509452700412448313739003075, −4.00716186536001279521923234189, −3.78769641466580568623112365695, −2.95181659001229895680726890061, −1.32962279936782404079285849890, 0,
1.32962279936782404079285849890, 2.95181659001229895680726890061, 3.78769641466580568623112365695, 4.00716186536001279521923234189, 5.76509452700412448313739003075, 6.68461495565388836489640760645, 7.23066919257894410409371315840, 8.119209499510440553100289543051, 8.632336773486661306730920493015
