L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s + 9-s − 3·10-s + 3·11-s + 12-s − 4·13-s + 3·15-s + 16-s − 18-s − 4·19-s + 3·20-s − 3·22-s − 24-s + 4·25-s + 4·26-s + 27-s + 9·29-s − 3·30-s − 31-s − 32-s + 3·33-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s + 0.288·12-s − 1.10·13-s + 0.774·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.639·22-s − 0.204·24-s + 4/5·25-s + 0.784·26-s + 0.192·27-s + 1.67·29-s − 0.547·30-s − 0.179·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.386085743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.386085743\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−11.73032787409280178574512521046, −10.41674905816924812365277509369, −9.784472360233863356102575379498, −9.103290655594877369002903439998, −8.157887841004923802013523625616, −6.89020910856782814907551199051, −6.11592597660017608535560188268, −4.63382915978845866280177166182, −2.81281773147692177384013154040, −1.68706416051862692239242100457,
1.68706416051862692239242100457, 2.81281773147692177384013154040, 4.63382915978845866280177166182, 6.11592597660017608535560188268, 6.89020910856782814907551199051, 8.157887841004923802013523625616, 9.103290655594877369002903439998, 9.784472360233863356102575379498, 10.41674905816924812365277509369, 11.73032787409280178574512521046