| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s − 11-s + 12-s + 2·14-s + 16-s − 17-s − 18-s + 8·19-s − 2·21-s + 22-s + 6·23-s − 24-s − 5·25-s + 27-s − 2·28-s − 2·29-s − 4·31-s − 32-s − 33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s + 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.902148851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902148851\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.20593281430673, −12.72244052465787, −12.07886124762810, −11.62820550468337, −11.26754191061994, −10.64439699984014, −10.05599107247932, −9.784352442675670, −9.287589843741546, −9.011965949752289, −8.310771530287095, −7.925285608732710, −7.384441876240494, −6.921377693699142, −6.644721748888535, −5.720713171890277, −5.446001825357712, −4.819292343194068, −3.980224991446422, −3.397814807721424, −3.121643822670922, −2.424607881936058, −1.854964729444039, −1.095718077700769, −0.4637225648789732,
0.4637225648789732, 1.095718077700769, 1.854964729444039, 2.424607881936058, 3.121643822670922, 3.397814807721424, 3.980224991446422, 4.819292343194068, 5.446001825357712, 5.720713171890277, 6.644721748888535, 6.921377693699142, 7.384441876240494, 7.925285608732710, 8.310771530287095, 9.011965949752289, 9.287589843741546, 9.784352442675670, 10.05599107247932, 10.64439699984014, 11.26754191061994, 11.62820550468337, 12.07886124762810, 12.72244052465787, 13.20593281430673
