L(s) = 1 | − 3-s + 7-s + 9-s − 11-s − 3·17-s + 3·19-s − 21-s − 23-s − 27-s − 7·29-s − 6·31-s + 33-s + 8·37-s − 2·41-s − 5·43-s − 6·47-s + 49-s + 3·51-s + 3·53-s − 3·57-s + 5·59-s − 7·61-s + 63-s + 6·67-s + 69-s + 10·71-s − 12·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.727·17-s + 0.688·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 1.29·29-s − 1.07·31-s + 0.174·33-s + 1.31·37-s − 0.312·41-s − 0.762·43-s − 0.875·47-s + 1/7·49-s + 0.420·51-s + 0.412·53-s − 0.397·57-s + 0.650·59-s − 0.896·61-s + 0.125·63-s + 0.733·67-s + 0.120·69-s + 1.18·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.01076463300152, −13.51528121057411, −13.07234786689018, −12.66291210943949, −12.07109086055896, −11.46791339981927, −11.21000106850233, −10.80386604577532, −10.11870276606852, −9.642006538184578, −9.198465155481313, −8.575823994392125, −7.954418139319111, −7.539031278892018, −6.985640090888644, −6.447389502010990, −5.818186430346925, −5.340481105239293, −4.888017713961826, −4.234415502008933, −3.674629267825101, −3.010745535185545, −2.136305406191786, −1.705519583411861, −0.7912134512306486, 0,
0.7912134512306486, 1.705519583411861, 2.136305406191786, 3.010745535185545, 3.674629267825101, 4.234415502008933, 4.888017713961826, 5.340481105239293, 5.818186430346925, 6.447389502010990, 6.985640090888644, 7.539031278892018, 7.954418139319111, 8.575823994392125, 9.198465155481313, 9.642006538184578, 10.11870276606852, 10.80386604577532, 11.21000106850233, 11.46791339981927, 12.07109086055896, 12.66291210943949, 13.07234786689018, 13.51528121057411, 14.01076463300152