L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 4·7-s + 8-s + 9-s − 10-s + 4·11-s + 12-s − 2·13-s − 4·14-s − 15-s + 16-s + 2·17-s + 18-s + 19-s − 20-s − 4·21-s + 4·22-s + 24-s + 25-s − 2·26-s + 27-s − 4·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.872·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.89229653121067, −12.42740077791680, −12.17799735094973, −11.66376104848947, −11.31234873620482, −10.56086460103999, −10.02943668547887, −9.675593724687759, −9.444031131358637, −8.742381686384185, −8.214265370024747, −7.835557088430538, −7.132152706349562, −6.732362527065195, −6.439950811489636, −6.019625631894769, −5.194745332663026, −4.784300931102437, −4.063783908328108, −3.807932252045158, −3.233048899715346, −2.825253975136840, −2.381718701605234, −1.420427790903841, −0.9131811422936304, 0,
0.9131811422936304, 1.420427790903841, 2.381718701605234, 2.825253975136840, 3.233048899715346, 3.807932252045158, 4.063783908328108, 4.784300931102437, 5.194745332663026, 6.019625631894769, 6.439950811489636, 6.732362527065195, 7.132152706349562, 7.835557088430538, 8.214265370024747, 8.742381686384185, 9.444031131358637, 9.675593724687759, 10.02943668547887, 10.56086460103999, 11.31234873620482, 11.66376104848947, 12.17799735094973, 12.42740077791680, 12.89229653121067