L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 2·11-s − 12-s − 6·13-s − 2·14-s + 16-s − 2·17-s − 18-s − 2·21-s − 2·22-s + 4·23-s + 24-s + 6·26-s − 27-s + 2·28-s − 8·31-s − 32-s − 2·33-s + 2·34-s + 36-s + 2·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 1.66·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.436·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s + 0.377·28-s − 1.43·31-s − 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.92958938520162, −12.36976420261922, −11.97558295376393, −11.41791276125142, −11.30320688933109, −10.60967701653439, −10.34591580095515, −9.715458181472110, −9.198232251684813, −9.048370346687625, −8.343726387077560, −7.724490348292984, −7.446397494176847, −6.915513882766363, −6.622220716186752, −5.794277784058832, −5.490688812295294, −4.748786484106904, −4.597783617579237, −3.818913219071443, −3.171778929224621, −2.391200792805686, −2.034718859012842, −1.363468771223671, −0.6993188931764371, 0,
0.6993188931764371, 1.363468771223671, 2.034718859012842, 2.391200792805686, 3.171778929224621, 3.818913219071443, 4.597783617579237, 4.748786484106904, 5.490688812295294, 5.794277784058832, 6.622220716186752, 6.915513882766363, 7.446397494176847, 7.724490348292984, 8.343726387077560, 9.048370346687625, 9.198232251684813, 9.715458181472110, 10.34591580095515, 10.60967701653439, 11.30320688933109, 11.41791276125142, 11.97558295376393, 12.36976420261922, 12.92958938520162