| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 8·7-s + 8-s + 9-s + 10-s + 12-s − 8·14-s + 15-s + 16-s − 12·17-s + 18-s + 20-s − 8·21-s + 24-s + 25-s + 27-s − 8·28-s + 30-s + 32-s − 12·34-s − 8·35-s + 36-s + 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 3.02·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 2.13·14-s + 0.258·15-s + 1/4·16-s − 2.91·17-s + 0.235·18-s + 0.223·20-s − 1.74·21-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 1.51·28-s + 0.182·30-s + 0.176·32-s − 2.05·34-s − 1.35·35-s + 1/6·36-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.276841907117526756164869085518, −8.811879039450229959581280002555, −8.644229019078172602684577487278, −7.53023481714424969283045906095, −6.97121259233254998318201926571, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −6.10897773314734163440451778623, −5.03186235211519720784151262088, −4.32687879913691587957737883725, −3.82839635062702052518767893213, −3.10714236804866821433728477742, −2.74628747875934588873509462774, −2.01462355044529552405323097963, 0,
2.01462355044529552405323097963, 2.74628747875934588873509462774, 3.10714236804866821433728477742, 3.82839635062702052518767893213, 4.32687879913691587957737883725, 5.03186235211519720784151262088, 6.10897773314734163440451778623, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 6.97121259233254998318201926571, 7.53023481714424969283045906095, 8.644229019078172602684577487278, 8.811879039450229959581280002555, 9.276841907117526756164869085518
