| L(s) = 1 | + 5-s + 7-s − 3·9-s − 11-s − 6·13-s + 6·17-s + 4·19-s + 8·23-s + 25-s − 10·29-s + 4·31-s + 35-s + 6·37-s − 10·41-s − 4·43-s − 3·45-s + 4·47-s + 49-s + 6·53-s − 55-s − 6·61-s − 3·63-s − 6·65-s − 4·67-s + 6·73-s − 77-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s − 0.301·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 0.447·45-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.768·61-s − 0.377·63-s − 0.744·65-s − 0.488·67-s + 0.702·73-s − 0.113·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.897151267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.897151267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.83402636086941902879205716806, −7.53301022574652468102829010620, −6.72374723465584942961574232026, −5.61012614671005261568067100455, −5.33992350829704284084310378069, −4.70265465766904474538029121449, −3.35114969031801505229541631378, −2.85279304071672136576456750833, −1.91837669920689412638048820619, −0.71169980966855863186211225278,
0.71169980966855863186211225278, 1.91837669920689412638048820619, 2.85279304071672136576456750833, 3.35114969031801505229541631378, 4.70265465766904474538029121449, 5.33992350829704284084310378069, 5.61012614671005261568067100455, 6.72374723465584942961574232026, 7.53301022574652468102829010620, 7.83402636086941902879205716806
