| L(s) = 1 | − 3-s − 5-s + 9-s + 11-s + 13-s + 15-s + 6·17-s + 4·19-s + 25-s − 27-s − 6·29-s + 4·31-s − 33-s − 2·37-s − 39-s + 2·41-s − 45-s − 7·49-s − 6·51-s + 6·53-s − 55-s − 4·57-s + 4·59-s + 14·61-s − 65-s + 8·67-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s − 0.328·37-s − 0.160·39-s + 0.312·41-s − 0.149·45-s − 49-s − 0.840·51-s + 0.824·53-s − 0.134·55-s − 0.529·57-s + 0.520·59-s + 1.79·61-s − 0.124·65-s + 0.977·67-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.832736605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.832736605\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.96791555090295, −15.43337215699598, −14.65709659626454, −14.40550197863365, −13.58707514824235, −13.08873452150391, −12.39231950569180, −11.90887218222898, −11.53953176973683, −10.89815464590536, −10.30442788489134, −9.623090370895380, −9.284913252968398, −8.182550093376329, −7.985369084228048, −7.113851933128593, −6.715593738738971, −5.785052755702582, −5.403232376370712, −4.725077180747497, −3.741187007488426, −3.488471119571877, −2.409746409654865, −1.337442275792379, −0.6644645629319405,
0.6644645629319405, 1.337442275792379, 2.409746409654865, 3.488471119571877, 3.741187007488426, 4.725077180747497, 5.403232376370712, 5.785052755702582, 6.715593738738971, 7.113851933128593, 7.985369084228048, 8.182550093376329, 9.284913252968398, 9.623090370895380, 10.30442788489134, 10.89815464590536, 11.53953176973683, 11.90887218222898, 12.39231950569180, 13.08873452150391, 13.58707514824235, 14.40550197863365, 14.65709659626454, 15.43337215699598, 15.96791555090295
