| L(s) = 1 | + 2·4-s + 9-s + 16-s + 2·19-s + 4·31-s + 2·36-s − 2·61-s − 2·64-s + 4·76-s − 2·79-s − 2·109-s − 2·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 2·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2·4-s + 9-s + 16-s + 2·19-s + 4·31-s + 2·36-s − 2·61-s − 2·64-s + 4·76-s − 2·79-s − 2·109-s − 2·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 2·171-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.968214201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.968214201\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.19194121835197976168501001974, −6.12057600771092268005644362242, −5.86761196510298996468311499597, −5.86554114594662525696287993542, −5.52187759384646165642743226718, −5.07587083301006102929593675467, −5.02764445749980491801074010514, −4.76888376393792232929542330680, −4.75699986730134726110633825330, −4.44999246186176198179348033402, −4.23451733100877893075091427592, −3.91130683483380075218868320430, −3.77701259269339771140015573393, −3.56070869084224192559535995863, −3.12309477230480353186186077405, −2.99317424055558212055139518912, −2.73657932803332693568753137616, −2.64389787921711101402887745290, −2.52015724114339707443049677301, −2.26877293931879172798436840643, −1.71567785294318915552831204670, −1.42156983889669302031478567055, −1.41504440321958248405963479284, −1.23400903229758329799400418931, −0.66246478570059125016810864412,
0.66246478570059125016810864412, 1.23400903229758329799400418931, 1.41504440321958248405963479284, 1.42156983889669302031478567055, 1.71567785294318915552831204670, 2.26877293931879172798436840643, 2.52015724114339707443049677301, 2.64389787921711101402887745290, 2.73657932803332693568753137616, 2.99317424055558212055139518912, 3.12309477230480353186186077405, 3.56070869084224192559535995863, 3.77701259269339771140015573393, 3.91130683483380075218868320430, 4.23451733100877893075091427592, 4.44999246186176198179348033402, 4.75699986730134726110633825330, 4.76888376393792232929542330680, 5.02764445749980491801074010514, 5.07587083301006102929593675467, 5.52187759384646165642743226718, 5.86554114594662525696287993542, 5.86761196510298996468311499597, 6.12057600771092268005644362242, 6.19194121835197976168501001974
