| L(s) = 1 | − 2·3-s − 4·7-s + 9-s − 4·13-s + 4·17-s − 4·19-s + 8·21-s − 6·23-s + 4·27-s − 10·29-s + 4·31-s + 2·37-s + 8·39-s + 10·41-s + 8·43-s + 6·47-s + 9·49-s − 8·51-s + 2·53-s + 8·57-s + 4·59-s − 10·61-s − 4·63-s − 2·67-s + 12·69-s + 12·73-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s + 0.970·17-s − 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s − 1.85·29-s + 0.718·31-s + 0.328·37-s + 1.28·39-s + 1.56·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s − 1.12·51-s + 0.274·53-s + 1.05·57-s + 0.520·59-s − 1.28·61-s − 0.503·63-s − 0.244·67-s + 1.44·69-s + 1.40·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.09095652375622, −12.60208014488503, −12.46833740837042, −12.04506804883130, −11.50745524193179, −10.92359053065024, −10.56756648182477, −9.976040837365664, −9.724950707038819, −9.233657142545142, −8.713202499623408, −7.817049528892021, −7.582541262433336, −6.973509652983598, −6.429885611878373, −5.958124144424759, −5.730692312957941, −5.216175166507378, −4.383041852361826, −4.043487830908272, −3.426374819739226, −2.574897614534365, −2.402125360799003, −1.303301648932222, −0.4970221426565379, 0,
0.4970221426565379, 1.303301648932222, 2.402125360799003, 2.574897614534365, 3.426374819739226, 4.043487830908272, 4.383041852361826, 5.216175166507378, 5.730692312957941, 5.958124144424759, 6.429885611878373, 6.973509652983598, 7.582541262433336, 7.817049528892021, 8.713202499623408, 9.233657142545142, 9.724950707038819, 9.976040837365664, 10.56756648182477, 10.92359053065024, 11.50745524193179, 12.04506804883130, 12.46833740837042, 12.60208014488503, 13.09095652375622
