L(s) = 1 | − 2·3-s + 3·7-s + 9-s − 5·11-s − 5·13-s − 4·17-s − 19-s − 6·21-s + 23-s + 4·27-s + 9·29-s + 2·31-s + 10·33-s − 2·37-s + 10·39-s + 3·41-s + 7·43-s + 12·47-s + 2·49-s + 8·51-s + 12·53-s + 2·57-s + 6·59-s − 10·61-s + 3·63-s + 8·67-s − 2·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s − 1.50·11-s − 1.38·13-s − 0.970·17-s − 0.229·19-s − 1.30·21-s + 0.208·23-s + 0.769·27-s + 1.67·29-s + 0.359·31-s + 1.74·33-s − 0.328·37-s + 1.60·39-s + 0.468·41-s + 1.06·43-s + 1.75·47-s + 2/7·49-s + 1.12·51-s + 1.64·53-s + 0.264·57-s + 0.781·59-s − 1.28·61-s + 0.377·63-s + 0.977·67-s − 0.240·69-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 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.23367419961639434976874347386, −6.79605833897650723426332827968, −5.73793861966768472871974303382, −5.35263379179179282869941939328, −4.68535313158255414533545755257, −4.32015692814401318756579777963, −2.64934542346709469981154920638, −2.38339433704264829829691281491, −0.972462805090003783209597895285, 0,
0.972462805090003783209597895285, 2.38339433704264829829691281491, 2.64934542346709469981154920638, 4.32015692814401318756579777963, 4.68535313158255414533545755257, 5.35263379179179282869941939328, 5.73793861966768472871974303382, 6.79605833897650723426332827968, 7.23367419961639434976874347386