L(s) = 1 | − 3-s + 9-s − 11-s − 5·13-s − 4·17-s − 5·19-s + 3·23-s − 27-s + 8·29-s − 6·31-s + 33-s − 9·37-s + 5·39-s + 3·41-s + 4·43-s + 47-s + 4·51-s + 9·53-s + 5·57-s + 4·59-s + 4·61-s − 4·67-s − 3·69-s + 2·71-s − 14·73-s − 4·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.970·17-s − 1.14·19-s + 0.625·23-s − 0.192·27-s + 1.48·29-s − 1.07·31-s + 0.174·33-s − 1.47·37-s + 0.800·39-s + 0.468·41-s + 0.609·43-s + 0.145·47-s + 0.560·51-s + 1.23·53-s + 0.662·57-s + 0.520·59-s + 0.512·61-s − 0.488·67-s − 0.361·69-s + 0.237·71-s − 1.63·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6106322491\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6106322491\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.11420571661955, −14.72411150029692, −14.19090297948654, −13.42245366240953, −13.00939080275638, −12.41726574316656, −12.06508013060354, −11.44339312440720, −10.68647956145802, −10.52805806999190, −9.875730152762739, −9.144248826505335, −8.696070379880743, −8.057827295388304, −7.154282390482639, −6.999522336388667, −6.319328386308044, −5.507844080127933, −5.078102259185937, −4.407510013350851, −3.926961189319924, −2.764050346819582, −2.372044448272856, −1.449626567147284, −0.3074860697255823,
0.3074860697255823, 1.449626567147284, 2.372044448272856, 2.764050346819582, 3.926961189319924, 4.407510013350851, 5.078102259185937, 5.507844080127933, 6.319328386308044, 6.999522336388667, 7.154282390482639, 8.057827295388304, 8.696070379880743, 9.144248826505335, 9.875730152762739, 10.52805806999190, 10.68647956145802, 11.44339312440720, 12.06508013060354, 12.41726574316656, 13.00939080275638, 13.42245366240953, 14.19090297948654, 14.72411150029692, 15.11420571661955