| L(s) = 1 | + 7·4-s − 9·9-s + 33·16-s + 44·19-s + 4·31-s − 63·36-s − 98·49-s − 236·61-s + 119·64-s + 308·76-s − 196·79-s + 81·81-s + 44·109-s + 242·121-s + 28·124-s + 127-s + 131-s + 137-s + 139-s − 297·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s − 396·171-s + ⋯ |
| L(s) = 1 | + 7/4·4-s − 9-s + 2.06·16-s + 2.31·19-s + 4/31·31-s − 7/4·36-s − 2·49-s − 3.86·61-s + 1.85·64-s + 4.05·76-s − 2.48·79-s + 81-s + 0.403·109-s + 2·121-s + 7/31·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.06·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s − 2.31·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.031955127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.031955127\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 382 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 98 T^{2} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4222 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1778 T^{2} + p^{4} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 118 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 98 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9938 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−14.76490280518785353993399146950, −13.92539311789386162852678130969, −13.76174265293518404568626967360, −12.76615296458415609702033283717, −12.11488173386944239510857561232, −11.83627455434379745526853383829, −11.18099988496288200452632945387, −11.04733130159140469325977840695, −10.15435910495107794465930274154, −9.638918405149932052416966794377, −8.879540449514900323300420046373, −7.995983009523014210497531158432, −7.55021665442649609824048918315, −6.98511199075596122388710643050, −6.12509675385144356007235067614, −5.75659578705248689833887837137, −4.84321135567696772073107231735, −3.23820338775733608420604808770, −2.92802658174998313751160576561, −1.57143054738411721911180195206,
1.57143054738411721911180195206, 2.92802658174998313751160576561, 3.23820338775733608420604808770, 4.84321135567696772073107231735, 5.75659578705248689833887837137, 6.12509675385144356007235067614, 6.98511199075596122388710643050, 7.55021665442649609824048918315, 7.995983009523014210497531158432, 8.879540449514900323300420046373, 9.638918405149932052416966794377, 10.15435910495107794465930274154, 11.04733130159140469325977840695, 11.18099988496288200452632945387, 11.83627455434379745526853383829, 12.11488173386944239510857561232, 12.76615296458415609702033283717, 13.76174265293518404568626967360, 13.92539311789386162852678130969, 14.76490280518785353993399146950
