| L(s) = 1 | − 4·3-s − 2·4-s + 4·7-s + 6·9-s + 8·12-s − 16·13-s − 5·16-s − 16·21-s + 2·25-s + 4·27-s − 8·28-s − 12·36-s + 64·39-s + 20·48-s − 2·49-s + 32·52-s + 24·63-s + 20·64-s + 32·73-s − 8·75-s + 32·79-s − 37·81-s + 32·84-s − 64·91-s + 32·97-s − 4·100-s − 40·103-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 4-s + 1.51·7-s + 2·9-s + 2.30·12-s − 4.43·13-s − 5/4·16-s − 3.49·21-s + 2/5·25-s + 0.769·27-s − 1.51·28-s − 2·36-s + 10.2·39-s + 2.88·48-s − 2/7·49-s + 4.43·52-s + 3.02·63-s + 5/2·64-s + 3.74·73-s − 0.923·75-s + 3.60·79-s − 4.11·81-s + 3.49·84-s − 6.70·91-s + 3.24·97-s − 2/5·100-s − 3.94·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1668046606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1668046606\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−10.19070171703711611628078833786, −9.813898458048051618481733890212, −9.690602564704510043371776154013, −9.480220212427536660710977679283, −9.254308968911655758612174884459, −8.787811672601936250810851613259, −8.336372474437866384699699670877, −8.187704968624703758502995805225, −7.64916586128995768211890223380, −7.61366997621558420853803955185, −7.16447954574929738174630481549, −6.68691565206057378813171168210, −6.60881536426890603723315013468, −6.40265755981170127584780735458, −5.57778899238268233473338911925, −5.22817115143474365843978716757, −5.11886651210099380061908742700, −4.92041555918160653379701026710, −4.84035481314464242271184792663, −4.38625364227784555644031732241, −4.01762483810657144674943936476, −3.01381048990394622851526182839, −2.34515833163741102245773879332, −2.13235648433281233133195991540, −0.47956781225255046972740175522,
0.47956781225255046972740175522, 2.13235648433281233133195991540, 2.34515833163741102245773879332, 3.01381048990394622851526182839, 4.01762483810657144674943936476, 4.38625364227784555644031732241, 4.84035481314464242271184792663, 4.92041555918160653379701026710, 5.11886651210099380061908742700, 5.22817115143474365843978716757, 5.57778899238268233473338911925, 6.40265755981170127584780735458, 6.60881536426890603723315013468, 6.68691565206057378813171168210, 7.16447954574929738174630481549, 7.61366997621558420853803955185, 7.64916586128995768211890223380, 8.187704968624703758502995805225, 8.336372474437866384699699670877, 8.787811672601936250810851613259, 9.254308968911655758612174884459, 9.480220212427536660710977679283, 9.690602564704510043371776154013, 9.813898458048051618481733890212, 10.19070171703711611628078833786
