| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 2·13-s − 14-s + 16-s − 6·17-s − 18-s + 21-s − 23-s − 24-s − 5·25-s + 2·26-s + 27-s + 28-s + 6·29-s − 8·31-s − 32-s + 6·34-s + 36-s + 6·37-s − 2·39-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 + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.986·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.95305576986278, −12.17901261274839, −11.73361531635488, −11.47858837345999, −10.90272926150349, −10.37795149643493, −10.03603040806168, −9.565596664330943, −9.079349255442770, −8.641807332719914, −8.304050263602305, −7.808713699913052, −7.346219487691525, −6.844845595705444, −6.490775000592652, −5.883995291277703, −5.204407987270334, −4.774160669385924, −4.203695318886310, −3.644245646777700, −3.102459080762656, −2.364984631849064, −2.044909723705088, −1.581402209804436, −0.6750727228408207, 0,
0.6750727228408207, 1.581402209804436, 2.044909723705088, 2.364984631849064, 3.102459080762656, 3.644245646777700, 4.203695318886310, 4.774160669385924, 5.204407987270334, 5.883995291277703, 6.490775000592652, 6.844845595705444, 7.346219487691525, 7.808713699913052, 8.304050263602305, 8.641807332719914, 9.079349255442770, 9.565596664330943, 10.03603040806168, 10.37795149643493, 10.90272926150349, 11.47858837345999, 11.73361531635488, 12.17901261274839, 12.95305576986278
