L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 2·11-s + 4·13-s + 15-s − 2·17-s − 2·19-s + 21-s + 25-s − 27-s + 6·29-s − 2·31-s − 2·33-s + 35-s − 10·37-s − 4·39-s − 10·41-s − 12·43-s − 45-s − 8·47-s + 49-s + 2·51-s − 2·55-s + 2·57-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s − 0.485·17-s − 0.458·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s − 1.64·37-s − 0.640·39-s − 1.56·41-s − 1.82·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.269·55-s + 0.264·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.24058081525793, −12.58989196909870, −12.25942165147569, −11.74715759264394, −11.39293183675577, −10.95053228366671, −10.35098162470682, −10.13941627496028, −9.456261582613978, −8.863903862658255, −8.479024311294105, −8.173688877939478, −7.397938966958942, −6.720790230455751, −6.564122915744250, −6.240225565292530, −5.341733439114454, −5.031914777718809, −4.424566967519443, −3.794130284864137, −3.442704446876719, −2.913946624309267, −1.809213557276801, −1.591249957748735, −0.6623149391226936, 0,
0.6623149391226936, 1.591249957748735, 1.809213557276801, 2.913946624309267, 3.442704446876719, 3.794130284864137, 4.424566967519443, 5.031914777718809, 5.341733439114454, 6.240225565292530, 6.564122915744250, 6.720790230455751, 7.397938966958942, 8.173688877939478, 8.479024311294105, 8.863903862658255, 9.456261582613978, 10.13941627496028, 10.35098162470682, 10.95053228366671, 11.39293183675577, 11.74715759264394, 12.25942165147569, 12.58989196909870, 13.24058081525793