| L(s) = 1 | − 4·2-s + 4·4-s + 16·8-s − 4·9-s − 64·16-s + 16·18-s + 20·25-s − 104·29-s + 64·32-s − 16·36-s − 8·37-s − 80·50-s − 184·53-s + 416·58-s + 192·64-s − 64·72-s + 32·74-s − 150·81-s + 80·100-s + 736·106-s + 376·109-s − 328·113-s − 416·116-s + 388·121-s + 127-s − 768·128-s + 131-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 4/9·9-s − 4·16-s + 8/9·18-s + 4/5·25-s − 3.58·29-s + 2·32-s − 4/9·36-s − 0.216·37-s − 8/5·50-s − 3.47·53-s + 7.17·58-s + 3·64-s − 8/9·72-s + 0.432·74-s − 1.85·81-s + 4/5·100-s + 6.94·106-s + 3.44·109-s − 2.90·113-s − 3.58·116-s + 3.20·121-s + 0.00787·127-s − 6·128-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.824978437\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.824978437\times10^{-5}\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2}( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 2 p T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 278 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 338 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 542 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 626 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1202 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 1202 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 1346 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 2462 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7382 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 674 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 4702 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 11714 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4958 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 12002 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 16658 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.950656883605508034947141144449, −8.806490240589052859842393300501, −8.380582587913998169033539494724, −8.213804249016889145484577740417, −7.86298309139532489896939700381, −7.64285990190553147089730117035, −7.56784251798072888310702512024, −7.09892192806918668236342965751, −6.87255087664357331854754358925, −6.74223925019524641949235061210, −6.10826419025100486717947337691, −5.74164017030225173110445804514, −5.67216954200315611974382408616, −5.25078197005026972351715664216, −4.73983162287860539323750546894, −4.54156840728168375730538618524, −4.40469107775862430902865258312, −3.71563548110425332905644664335, −3.41924863655201342891431368528, −3.18795497438030496656277540965, −2.32831178701531051690533968605, −1.95543324863101546537582547867, −1.57502775722375061344108483095, −1.01411373812635109549442986111, −0.00316819457135546899248946937,
0.00316819457135546899248946937, 1.01411373812635109549442986111, 1.57502775722375061344108483095, 1.95543324863101546537582547867, 2.32831178701531051690533968605, 3.18795497438030496656277540965, 3.41924863655201342891431368528, 3.71563548110425332905644664335, 4.40469107775862430902865258312, 4.54156840728168375730538618524, 4.73983162287860539323750546894, 5.25078197005026972351715664216, 5.67216954200315611974382408616, 5.74164017030225173110445804514, 6.10826419025100486717947337691, 6.74223925019524641949235061210, 6.87255087664357331854754358925, 7.09892192806918668236342965751, 7.56784251798072888310702512024, 7.64285990190553147089730117035, 7.86298309139532489896939700381, 8.213804249016889145484577740417, 8.380582587913998169033539494724, 8.806490240589052859842393300501, 8.950656883605508034947141144449
