L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s + 4·11-s + 12-s + 2·13-s + 2·15-s + 16-s + 17-s + 18-s − 4·19-s + 2·20-s + 4·22-s + 24-s − 25-s + 2·26-s + 27-s + 10·29-s + 2·30-s − 8·31-s + 32-s + 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.85·29-s + 0.365·30-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225318 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225318 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 47 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.26102953904186, −12.85101297197173, −12.34874665456452, −11.82750548032603, −11.47901535774931, −10.86735551439022, −10.29136433198821, −9.945847752354283, −9.560157710839117, −8.810862780741319, −8.521750350523922, −8.190391329659163, −7.262848178493098, −6.883443790751105, −6.318721272275547, −6.225580133811040, −5.287952993295670, −5.087498284776965, −4.266342303930077, −3.836946831592949, −3.403509332555201, −2.765242811096165, −2.142036052588897, −1.528644390739810, −1.288923601784756, 0,
1.288923601784756, 1.528644390739810, 2.142036052588897, 2.765242811096165, 3.403509332555201, 3.836946831592949, 4.266342303930077, 5.087498284776965, 5.287952993295670, 6.225580133811040, 6.318721272275547, 6.883443790751105, 7.262848178493098, 8.190391329659163, 8.521750350523922, 8.810862780741319, 9.560157710839117, 9.945847752354283, 10.29136433198821, 10.86735551439022, 11.47901535774931, 11.82750548032603, 12.34874665456452, 12.85101297197173, 13.26102953904186