L(s) = 1 | + 2·5-s − 7-s − 3·9-s − 2·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s − 2·37-s − 2·41-s − 4·43-s − 6·45-s + 8·47-s + 49-s + 6·53-s + 6·61-s + 3·63-s − 4·65-s + 4·67-s + 8·71-s − 10·73-s + 16·79-s + 9·81-s + 8·83-s + 12·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.768·61-s + 0.377·63-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 81-s + 0.878·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.109489904\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.109489904\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.55396780737299, −15.65107129434973, −14.93465069367051, −14.43804506389249, −13.93938499767119, −13.56207845299293, −12.83742592947578, −12.18105184557004, −11.74279248999077, −11.12047980717424, −10.32184855885086, −9.813761635026272, −9.360540366280823, −8.872019738763444, −7.882546856710184, −7.484410531599589, −6.768784368125564, −5.836485557210028, −5.449971086197904, −5.183783273263211, −3.748711321592123, −3.315325009962182, −2.501208561951002, −1.705643996975220, −0.6412279080455006,
0.6412279080455006, 1.705643996975220, 2.501208561951002, 3.315325009962182, 3.748711321592123, 5.183783273263211, 5.449971086197904, 5.836485557210028, 6.768784368125564, 7.484410531599589, 7.882546856710184, 8.872019738763444, 9.360540366280823, 9.813761635026272, 10.32184855885086, 11.12047980717424, 11.74279248999077, 12.18105184557004, 12.83742592947578, 13.56207845299293, 13.93938499767119, 14.43804506389249, 14.93465069367051, 15.65107129434973, 16.55396780737299