L(s) = 1 | + 2·3-s + 3·7-s + 9-s − 6·13-s − 7·17-s + 4·19-s + 6·21-s + 23-s − 4·27-s − 9·29-s + 3·31-s − 7·37-s − 12·39-s + 9·41-s + 4·43-s + 2·47-s + 2·49-s − 14·51-s − 7·53-s + 8·57-s − 9·59-s − 2·61-s + 3·63-s − 13·67-s + 2·69-s + 13·71-s − 4·73-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.13·7-s + 1/3·9-s − 1.66·13-s − 1.69·17-s + 0.917·19-s + 1.30·21-s + 0.208·23-s − 0.769·27-s − 1.67·29-s + 0.538·31-s − 1.15·37-s − 1.92·39-s + 1.40·41-s + 0.609·43-s + 0.291·47-s + 2/7·49-s − 1.96·51-s − 0.961·53-s + 1.05·57-s − 1.17·59-s − 0.256·61-s + 0.377·63-s − 1.58·67-s + 0.240·69-s + 1.54·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.52246097639366876208254473435, −7.05498017301087873222996716119, −5.96415214501196404120354284803, −5.09117471152249895096623105734, −4.60512350573298215100320611453, −3.82689952300772687711030084359, −2.83183807701483073477843055711, −2.28457606314058353081220807317, −1.59013445025816081356170839496, 0,
1.59013445025816081356170839496, 2.28457606314058353081220807317, 2.83183807701483073477843055711, 3.82689952300772687711030084359, 4.60512350573298215100320611453, 5.09117471152249895096623105734, 5.96415214501196404120354284803, 7.05498017301087873222996716119, 7.52246097639366876208254473435