| L(s) = 1 | − 9-s + 2·25-s − 2·49-s + 4·73-s + 81-s + 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2·225-s + ⋯ |
| L(s) = 1 | − 9-s + 2·25-s − 2·49-s + 4·73-s + 81-s + 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8249252067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8249252067\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$ | \( ( 1 - T )^{4} \) |
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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
−10.58411748633036289322809683421, −10.57579604182470278989178742714, −9.842610997450187228092650115146, −9.309281712207652705462010343223, −9.183908354581915427311547196559, −8.503231277316214331094544444154, −8.296824494016569845828844685206, −7.83229911155073064036663204386, −7.29631093228283246496110921738, −6.74423643649784553539034866123, −6.35265223941422036836609645592, −6.04541178656737113000583711432, −5.18517096887695521932682548074, −5.04220861762749904743178176906, −4.56634549720434133804464441339, −3.56565119294518014715373598869, −3.41007626517704594129377904426, −2.63984763760398080177827851398, −2.11411251405953769947321261844, −1.02777572148403383408704263892,
1.02777572148403383408704263892, 2.11411251405953769947321261844, 2.63984763760398080177827851398, 3.41007626517704594129377904426, 3.56565119294518014715373598869, 4.56634549720434133804464441339, 5.04220861762749904743178176906, 5.18517096887695521932682548074, 6.04541178656737113000583711432, 6.35265223941422036836609645592, 6.74423643649784553539034866123, 7.29631093228283246496110921738, 7.83229911155073064036663204386, 8.296824494016569845828844685206, 8.503231277316214331094544444154, 9.183908354581915427311547196559, 9.309281712207652705462010343223, 9.842610997450187228092650115146, 10.57579604182470278989178742714, 10.58411748633036289322809683421
