L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 12-s − 4·13-s − 14-s + 16-s + 18-s + 2·19-s + 21-s + 9·23-s − 24-s − 4·26-s − 27-s − 28-s − 2·31-s + 32-s + 36-s + 7·37-s + 2·38-s + 4·39-s − 3·41-s + 42-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 − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.218·21-s + 1.87·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.359·31-s + 0.176·32-s + 1/6·36-s + 1.15·37-s + 0.324·38-s + 0.640·39-s − 0.468·41-s + 0.154·42-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.74273684892944, −12.68003339771013, −11.86390168307007, −11.74177317825198, −11.06732948056748, −10.86304697547110, −10.18205249414136, −9.738058772781786, −9.386935576322124, −8.882528415418147, −7.978672304594026, −7.845157851161692, −7.022705298945039, −6.759384680000894, −6.499787948756623, −5.578456501627772, −5.354430556956488, −4.921863802638904, −4.401419356524310, −3.849483096858263, −3.111207576514727, −2.864837166131039, −2.153622008140987, −1.438889167038556, −0.7934489309612372, 0,
0.7934489309612372, 1.438889167038556, 2.153622008140987, 2.864837166131039, 3.111207576514727, 3.849483096858263, 4.401419356524310, 4.921863802638904, 5.354430556956488, 5.578456501627772, 6.499787948756623, 6.759384680000894, 7.022705298945039, 7.845157851161692, 7.978672304594026, 8.882528415418147, 9.386935576322124, 9.738058772781786, 10.18205249414136, 10.86304697547110, 11.06732948056748, 11.74177317825198, 11.86390168307007, 12.68003339771013, 12.74273684892944