L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 4·11-s + 2·13-s − 14-s − 16-s − 6·17-s + 4·19-s + 4·22-s − 2·26-s − 28-s + 2·29-s − 5·32-s + 6·34-s − 6·37-s − 4·38-s − 2·41-s + 4·43-s + 4·44-s + 49-s − 2·52-s + 6·53-s + 3·56-s − 2·58-s − 12·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.20·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.852·22-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 1/7·49-s − 0.277·52-s + 0.824·53-s + 0.400·56-s − 0.262·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.941556964198447272384176621072, −8.352294990335304362757323779032, −7.65494363024181651773191318169, −6.85360812870284917864517668882, −5.61533530390354367379245545667, −4.87983646890892800864940889103, −4.03796996879678948025961788837, −2.72775334283054769167882483900, −1.46036150187712502072371434143, 0,
1.46036150187712502072371434143, 2.72775334283054769167882483900, 4.03796996879678948025961788837, 4.87983646890892800864940889103, 5.61533530390354367379245545667, 6.85360812870284917864517668882, 7.65494363024181651773191318169, 8.352294990335304362757323779032, 8.941556964198447272384176621072