L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 8-s − 2·9-s − 3·10-s − 3·11-s − 12-s + 13-s − 3·15-s + 16-s + 2·18-s + 4·19-s + 3·20-s + 3·22-s − 6·23-s + 24-s + 4·25-s − 26-s + 5·27-s + 29-s + 3·30-s − 5·31-s − 32-s + 3·33-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 − 2/3·9-s − 0.948·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.774·15-s + 1/4·16-s + 0.471·18-s + 0.917·19-s + 0.670·20-s + 0.639·22-s − 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.962·27-s + 0.185·29-s + 0.547·30-s − 0.898·31-s − 0.176·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.468996869393629986402269684915, −7.75736788007027639104744920068, −6.83540443308828863491965690327, −5.85651335858910882186046194043, −5.75104594381671316212249852798, −4.78932264948245754590082821986, −3.26851249660750246952308406904, −2.39322139200660777765397167401, −1.45132308349722540495901611770, 0,
1.45132308349722540495901611770, 2.39322139200660777765397167401, 3.26851249660750246952308406904, 4.78932264948245754590082821986, 5.75104594381671316212249852798, 5.85651335858910882186046194043, 6.83540443308828863491965690327, 7.75736788007027639104744920068, 8.468996869393629986402269684915