| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s + 8·11-s − 4·16-s − 32·22-s + 4·25-s + 32·32-s + 24·43-s + 64·44-s − 2·49-s − 16·50-s − 64·64-s + 40·67-s − 96·86-s − 64·88-s + 8·98-s + 32·100-s + 40·107-s − 80·113-s − 4·121-s + 127-s + 64·128-s + 131-s − 160·134-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 2.82·8-s + 2.41·11-s − 16-s − 6.82·22-s + 4/5·25-s + 5.65·32-s + 3.65·43-s + 9.64·44-s − 2/7·49-s − 2.26·50-s − 8·64-s + 4.88·67-s − 10.3·86-s − 6.82·88-s + 0.808·98-s + 16/5·100-s + 3.86·107-s − 7.52·113-s − 0.363·121-s + 0.0887·127-s + 5.65·128-s + 0.0873·131-s − 13.8·134-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6978721546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6978721546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.196398448465406580953985062018, −7.58148818971757696102667669646, −7.53258055365310709124955844104, −7.32379461670124567664132011176, −7.03997022091557205765292905236, −6.76915033850979135925335320528, −6.67591707609765622212713724613, −6.36470307393936787382590872560, −6.20389807758513087279361355351, −5.88154691569736404913858123822, −5.38380406918337699017237087153, −5.30049183317735059237189828760, −4.75337781209501521388492084182, −4.58013551054993211992679579359, −4.26034556935623912322598867249, −3.92564134664087775194198994523, −3.62779921343568044316781763877, −3.59817122738588803860458302893, −2.67980867397175825308727844898, −2.46995466914332457304949869819, −2.30565822536694976416173352303, −1.70383546586965223269484873836, −1.18378488382907941152311842077, −1.11892292735081616892222788900, −0.58384853917214874937567409568,
0.58384853917214874937567409568, 1.11892292735081616892222788900, 1.18378488382907941152311842077, 1.70383546586965223269484873836, 2.30565822536694976416173352303, 2.46995466914332457304949869819, 2.67980867397175825308727844898, 3.59817122738588803860458302893, 3.62779921343568044316781763877, 3.92564134664087775194198994523, 4.26034556935623912322598867249, 4.58013551054993211992679579359, 4.75337781209501521388492084182, 5.30049183317735059237189828760, 5.38380406918337699017237087153, 5.88154691569736404913858123822, 6.20389807758513087279361355351, 6.36470307393936787382590872560, 6.67591707609765622212713724613, 6.76915033850979135925335320528, 7.03997022091557205765292905236, 7.32379461670124567664132011176, 7.53258055365310709124955844104, 7.58148818971757696102667669646, 8.196398448465406580953985062018
