| L(s) = 1 | + 4·2-s − 14·3-s + 8·4-s − 56·6-s + 18·7-s + 98·9-s − 112·12-s + 72·14-s − 64·16-s + 392·18-s − 252·21-s − 134·23-s − 378·27-s + 144·28-s − 256·32-s + 784·36-s + 504·41-s − 1.00e3·42-s − 594·43-s − 536·46-s − 602·47-s + 896·48-s + 162·49-s − 1.51e3·54-s + 1.90e3·61-s + 1.76e3·63-s − 512·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.69·3-s + 4-s − 3.81·6-s + 0.971·7-s + 3.62·9-s − 2.69·12-s + 1.37·14-s − 16-s + 5.13·18-s − 2.61·21-s − 1.21·23-s − 2.69·27-s + 0.971·28-s − 1.41·32-s + 3.62·36-s + 1.91·41-s − 3.70·42-s − 2.10·43-s − 1.71·46-s − 1.86·47-s + 2.69·48-s + 0.472·49-s − 3.81·54-s + 3.99·61-s + 3.52·63-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.276067165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276067165\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 134 T + 8978 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 44858 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 252 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 594 T + 176418 T^{2} + 594 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 602 T + 181202 T^{2} + 602 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 952 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1098 T + 602802 T^{2} - 1098 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 154 T + 11858 T^{2} + 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 511058 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.44590142657255398849546386506, −12.78763722534897934842312396183, −12.66688392134246848714796445468, −11.78857507227965236633734971245, −11.75617439549943575532024330042, −11.12450373931333869378797220692, −11.12394367225585336853777149573, −10.10631663274852436384618469023, −9.793472451580803520103023046183, −8.559695737286069964769095331831, −7.946842586839339440999870455249, −6.77880358472183029510199926431, −6.68426342846807777780759160001, −5.73770292474127798679729874667, −5.59452302201055361261685849755, −4.90153310742985723173546508756, −4.51371616882221718046987822520, −3.63171088902555101361970585774, −2.04998539968507989442704800075, −0.60626184763115811491319053032,
0.60626184763115811491319053032, 2.04998539968507989442704800075, 3.63171088902555101361970585774, 4.51371616882221718046987822520, 4.90153310742985723173546508756, 5.59452302201055361261685849755, 5.73770292474127798679729874667, 6.68426342846807777780759160001, 6.77880358472183029510199926431, 7.946842586839339440999870455249, 8.559695737286069964769095331831, 9.793472451580803520103023046183, 10.10631663274852436384618469023, 11.12394367225585336853777149573, 11.12450373931333869378797220692, 11.75617439549943575532024330042, 11.78857507227965236633734971245, 12.66688392134246848714796445468, 12.78763722534897934842312396183, 14.44590142657255398849546386506
