L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 11-s − 12-s − 13-s + 14-s + 16-s − 18-s + 21-s + 22-s + 4·23-s + 24-s + 26-s − 27-s − 28-s − 10·29-s − 3·31-s − 32-s + 33-s + 36-s + 2·37-s + 39-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 − 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s − 0.538·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s + 0.328·37-s + 0.160·39-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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−12.82926526681731, −12.37437822588752, −12.03140649196275, −11.29679283501246, −10.99417314562596, −10.76390813095907, −10.19255065090136, −9.526534120346767, −9.326610081843890, −8.994969200471155, −8.214540783543288, −7.724135592702054, −7.339472718476186, −6.989438421215633, −6.318261393471559, −5.927251255680335, −5.399469389548061, −5.020630203018969, −4.219864502834417, −3.786683859559879, −3.161254103593102, −2.448442541507389, −2.085189473043908, −1.236061716404530, −0.6724641580173804, 0,
0.6724641580173804, 1.236061716404530, 2.085189473043908, 2.448442541507389, 3.161254103593102, 3.786683859559879, 4.219864502834417, 5.020630203018969, 5.399469389548061, 5.927251255680335, 6.318261393471559, 6.989438421215633, 7.339472718476186, 7.724135592702054, 8.214540783543288, 8.994969200471155, 9.326610081843890, 9.526534120346767, 10.19255065090136, 10.76390813095907, 10.99417314562596, 11.29679283501246, 12.03140649196275, 12.37437822588752, 12.82926526681731