| L(s) = 1 | + 2-s − 4-s − 2·5-s + 7-s − 3·8-s − 2·10-s − 2·13-s + 14-s − 16-s − 2·17-s − 4·19-s + 2·20-s − 25-s − 2·26-s − 28-s − 29-s + 5·32-s − 2·34-s − 2·35-s − 2·37-s − 4·38-s + 6·40-s − 2·41-s − 4·43-s + 8·47-s + 49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.185·29-s + 0.883·32-s − 0.342·34-s − 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.948·40-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9558139384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9558139384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.08000175423945, −12.41521217043620, −12.14592841254709, −11.75271556679261, −11.13665526177888, −10.88068678806426, −10.05878047180225, −9.807133316619157, −9.098716369070565, −8.574749796841212, −8.400174089115991, −7.749741975182805, −7.245722814193810, −6.744518072199599, −6.150942171476305, −5.628760118056957, −5.018556549534675, −4.690226850847422, −4.131237795924381, −3.738732501137804, −3.291413048095568, −2.400152547674074, −2.094466719415014, −1.000180511388631, −0.2772303777487793,
0.2772303777487793, 1.000180511388631, 2.094466719415014, 2.400152547674074, 3.291413048095568, 3.738732501137804, 4.131237795924381, 4.690226850847422, 5.018556549534675, 5.628760118056957, 6.150942171476305, 6.744518072199599, 7.245722814193810, 7.749741975182805, 8.400174089115991, 8.574749796841212, 9.098716369070565, 9.807133316619157, 10.05878047180225, 10.88068678806426, 11.13665526177888, 11.75271556679261, 12.14592841254709, 12.41521217043620, 13.08000175423945
