L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s + 9-s + 8·10-s + 3·11-s − 4·12-s − 5·13-s + 8·15-s − 4·16-s − 3·17-s − 2·18-s − 2·19-s − 8·20-s − 6·22-s − 23-s + 11·25-s + 10·26-s + 4·27-s − 6·29-s − 16·30-s − 31-s + 8·32-s − 6·33-s + 6·34-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s + 1/3·9-s + 2.52·10-s + 0.904·11-s − 1.15·12-s − 1.38·13-s + 2.06·15-s − 16-s − 0.727·17-s − 0.471·18-s − 0.458·19-s − 1.78·20-s − 1.27·22-s − 0.208·23-s + 11/5·25-s + 1.96·26-s + 0.769·27-s − 1.11·29-s − 2.92·30-s − 0.179·31-s + 1.41·32-s − 1.04·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.98022739565726202171464841322, −14.78796968128268794927127081145, −12.38343238787281691828884133629, −11.51353349230783479982128065021, −10.79718018756646717446356079583, −9.206795357679225432153664688284, −7.86442320164458175925978217803, −6.82871744579371622309755168674, −4.49472027398106904117777838547, 0,
4.49472027398106904117777838547, 6.82871744579371622309755168674, 7.86442320164458175925978217803, 9.206795357679225432153664688284, 10.79718018756646717446356079583, 11.51353349230783479982128065021, 12.38343238787281691828884133629, 14.78796968128268794927127081145, 15.98022739565726202171464841322