| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·7-s − 3·8-s + 9-s − 11-s − 12-s + 13-s − 3·14-s − 16-s − 5·17-s + 18-s − 8·19-s − 3·21-s − 22-s − 3·24-s + 26-s + 27-s + 3·28-s + 29-s + 3·31-s + 5·32-s − 33-s − 5·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.13·7-s − 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s − 1/4·16-s − 1.21·17-s + 0.235·18-s − 1.83·19-s − 0.654·21-s − 0.213·22-s − 0.612·24-s + 0.196·26-s + 0.192·27-s + 0.566·28-s + 0.185·29-s + 0.538·31-s + 0.883·32-s − 0.174·33-s − 0.857·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.486501770041011871405261786646, −8.782211705481132856616595687911, −8.155742963526271445866193321322, −6.66347145599998265817904262316, −6.30841861416020206560317566803, −5.00207112486752800575218163446, −4.13449620600701812929984654486, −3.33612069944952559164671650565, −2.30526455032619611611112680077, 0,
2.30526455032619611611112680077, 3.33612069944952559164671650565, 4.13449620600701812929984654486, 5.00207112486752800575218163446, 6.30841861416020206560317566803, 6.66347145599998265817904262316, 8.155742963526271445866193321322, 8.782211705481132856616595687911, 9.486501770041011871405261786646
