L(s) = 1 | − 3-s + 5-s + 9-s − 5·11-s − 15-s + 4·17-s + 8·19-s − 4·23-s − 4·25-s − 27-s + 5·29-s − 3·31-s + 5·33-s + 4·37-s − 2·43-s + 45-s + 6·47-s − 4·51-s + 9·53-s − 5·55-s − 8·57-s − 11·59-s − 6·61-s + 2·67-s + 4·69-s + 2·71-s − 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.50·11-s − 0.258·15-s + 0.970·17-s + 1.83·19-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 0.928·29-s − 0.538·31-s + 0.870·33-s + 0.657·37-s − 0.304·43-s + 0.149·45-s + 0.875·47-s − 0.560·51-s + 1.23·53-s − 0.674·55-s − 1.05·57-s − 1.43·59-s − 0.768·61-s + 0.244·67-s + 0.481·69-s + 0.237·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.564166087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.564166087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.66704391563267466408747033453, −7.15916233678585242896748487052, −6.07186208611442724160614588288, −5.63572265312026843734928392292, −5.17949058979060087276286358320, −4.34753614684461177561158872944, −3.33263639061551217851718650296, −2.65508443181212071696941839887, −1.65045674899787460442941380444, −0.62514295325464923881788964725,
0.62514295325464923881788964725, 1.65045674899787460442941380444, 2.65508443181212071696941839887, 3.33263639061551217851718650296, 4.34753614684461177561158872944, 5.17949058979060087276286358320, 5.63572265312026843734928392292, 6.07186208611442724160614588288, 7.15916233678585242896748487052, 7.66704391563267466408747033453