L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 3·11-s + 12-s + 13-s + 14-s + 16-s − 18-s − 4·19-s − 21-s + 3·22-s + 6·23-s − 24-s − 26-s + 27-s − 28-s − 8·29-s + 7·31-s − 32-s − 3·33-s + 36-s + 10·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.218·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.48·29-s + 1.25·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−12.95388698109769, −12.50721638310061, −12.01144266866643, −11.31985685708892, −11.02524706104596, −10.52485996347340, −10.14344591973864, −9.673973233091389, −9.005914878806618, −8.969985466946001, −8.296298786315874, −7.874476932724701, −7.358178782819505, −7.102741763324520, −6.326161785806057, −6.026407569991417, −5.399230413327914, −4.830683490597641, −4.117191580284923, −3.781892657367772, −2.968750119306308, −2.519566919748145, −2.279526947942561, −1.306211364549507, −0.8038914019793767, 0,
0.8038914019793767, 1.306211364549507, 2.279526947942561, 2.519566919748145, 2.968750119306308, 3.781892657367772, 4.117191580284923, 4.830683490597641, 5.399230413327914, 6.026407569991417, 6.326161785806057, 7.102741763324520, 7.358178782819505, 7.874476932724701, 8.296298786315874, 8.969985466946001, 9.005914878806618, 9.673973233091389, 10.14344591973864, 10.52485996347340, 11.02524706104596, 11.31985685708892, 12.01144266866643, 12.50721638310061, 12.95388698109769