| L(s) = 1 | − 2-s − 2·3-s − 4-s − 5-s + 2·6-s − 2·7-s + 3·8-s + 9-s + 10-s − 11-s + 2·12-s + 6·13-s + 2·14-s + 2·15-s − 16-s − 18-s − 19-s + 20-s + 4·21-s + 22-s − 6·24-s + 25-s − 6·26-s + 4·27-s + 2·28-s + 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s + 1.66·13-s + 0.534·14-s + 0.516·15-s − 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s + 0.213·22-s − 1.22·24-s + 1/5·25-s − 1.17·26-s + 0.769·27-s + 0.377·28-s + 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−9.581897899326145556759519845409, −8.572381866452314942168735585083, −8.144528575687806914758667967843, −6.83006102631047061488008161722, −6.22130311336718161147659456809, −5.22519121956999151938912253710, −4.30702204859644929662120519966, −3.23559981947008472823955378050, −1.17080042590956595747782018947, 0,
1.17080042590956595747782018947, 3.23559981947008472823955378050, 4.30702204859644929662120519966, 5.22519121956999151938912253710, 6.22130311336718161147659456809, 6.83006102631047061488008161722, 8.144528575687806914758667967843, 8.572381866452314942168735585083, 9.581897899326145556759519845409
