| L(s) = 1 | + 2-s + 2·3-s + 2·6-s + 2·7-s + 9-s + 2·14-s + 18-s + 4·21-s + 2·23-s − 2·29-s − 32-s + 3·41-s + 4·42-s + 2·43-s + 2·46-s + 2·47-s + 49-s − 2·58-s − 2·61-s + 2·63-s − 64-s + 2·67-s + 4·69-s + 3·82-s − 3·83-s + 2·86-s − 4·87-s + ⋯ |
| L(s) = 1 | + 2-s + 2·3-s + 2·6-s + 2·7-s + 9-s + 2·14-s + 18-s + 4·21-s + 2·23-s − 2·29-s − 32-s + 3·41-s + 4·42-s + 2·43-s + 2·46-s + 2·47-s + 49-s − 2·58-s − 2·61-s + 2·63-s − 64-s + 2·67-s + 4·69-s + 3·82-s − 3·83-s + 2·86-s − 4·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(7.025181663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.025181663\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 7 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 29 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 47 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−6.42213669012807378572691820264, −6.19810842539751469935907022040, −6.04249410243797351577465301196, −5.79360395593900857721083594973, −5.57457571373889726278445499898, −5.47647887680163473249673558240, −5.26913128329759864484970343098, −5.01359347097360965261668888613, −4.74010947108876759133897983603, −4.56780752508732923047161270625, −4.46203862343632856159862758270, −4.09924138041649695139441526880, −4.03920749211525859179218939814, −3.88636518146787945192765184401, −3.55155787620817657507054437706, −3.18877364614556452628517955038, −3.04189233844060612608802731490, −2.84018867329524142305940854648, −2.49823782775422098636233583311, −2.47492414731401688452165838901, −2.00617053881978415967273458275, −2.00222429803606701063238480600, −1.44181257772867618342750506408, −1.13390886438567413976883468019, −0.960225447932536745197182587749,
0.960225447932536745197182587749, 1.13390886438567413976883468019, 1.44181257772867618342750506408, 2.00222429803606701063238480600, 2.00617053881978415967273458275, 2.47492414731401688452165838901, 2.49823782775422098636233583311, 2.84018867329524142305940854648, 3.04189233844060612608802731490, 3.18877364614556452628517955038, 3.55155787620817657507054437706, 3.88636518146787945192765184401, 4.03920749211525859179218939814, 4.09924138041649695139441526880, 4.46203862343632856159862758270, 4.56780752508732923047161270625, 4.74010947108876759133897983603, 5.01359347097360965261668888613, 5.26913128329759864484970343098, 5.47647887680163473249673558240, 5.57457571373889726278445499898, 5.79360395593900857721083594973, 6.04249410243797351577465301196, 6.19810842539751469935907022040, 6.42213669012807378572691820264
