L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s + 11-s + 2·13-s + 14-s + 16-s + 2·17-s + 3·18-s − 2·19-s − 22-s + 4·23-s − 2·26-s − 28-s − 4·29-s − 8·31-s − 32-s − 2·34-s − 3·36-s + 2·37-s + 2·38-s + 4·43-s + 44-s − 4·46-s − 2·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.458·19-s − 0.213·22-s + 0.834·23-s − 0.392·26-s − 0.188·28-s − 0.742·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.328·37-s + 0.324·38-s + 0.609·43-s + 0.150·44-s − 0.589·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.315888151335656953967584946052, −7.39496672650267543655917848432, −6.79897107063585475184666370987, −5.85359136681916693493643462990, −5.46297461195598588724093631116, −4.10252544297103874145558903467, −3.28981707601855422396426973259, −2.44851373595584787121115394269, −1.28082721166073117199936666654, 0,
1.28082721166073117199936666654, 2.44851373595584787121115394269, 3.28981707601855422396426973259, 4.10252544297103874145558903467, 5.46297461195598588724093631116, 5.85359136681916693493643462990, 6.79897107063585475184666370987, 7.39496672650267543655917848432, 8.315888151335656953967584946052