L(s) = 1 | − 2-s + 4-s − 8-s − 5·11-s − 13-s + 16-s + 2·17-s − 2·19-s + 5·22-s + 8·23-s + 26-s + 5·29-s − 4·31-s − 32-s − 2·34-s + 10·37-s + 2·38-s − 9·41-s + 5·43-s − 5·44-s − 8·46-s − 2·47-s − 52-s + 9·53-s − 5·58-s + 9·59-s + 12·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.458·19-s + 1.06·22-s + 1.66·23-s + 0.196·26-s + 0.928·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 1.64·37-s + 0.324·38-s − 1.40·41-s + 0.762·43-s − 0.753·44-s − 1.17·46-s − 0.291·47-s − 0.138·52-s + 1.23·53-s − 0.656·58-s + 1.17·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.92934301415026, −12.68423284806687, −11.91254368748999, −11.37017575645322, −11.20761438898817, −10.50597933068388, −10.13029824159238, −9.939652703887917, −9.241841792687361, −8.706500770045200, −8.359523436702524, −7.942941721151811, −7.332433232191022, −7.016533156311362, −6.580631798118703, −5.725174780849550, −5.468371878664966, −4.972937229209954, −4.349843843734831, −3.720204785330786, −2.889612740621029, −2.714032463023212, −2.140898379117806, −1.277778831274931, −0.7381111629657207, 0,
0.7381111629657207, 1.277778831274931, 2.140898379117806, 2.714032463023212, 2.889612740621029, 3.720204785330786, 4.349843843734831, 4.972937229209954, 5.468371878664966, 5.725174780849550, 6.580631798118703, 7.016533156311362, 7.332433232191022, 7.942941721151811, 8.359523436702524, 8.706500770045200, 9.241841792687361, 9.939652703887917, 10.13029824159238, 10.50597933068388, 11.20761438898817, 11.37017575645322, 11.91254368748999, 12.68423284806687, 12.92934301415026