| L(s) = 1 | + 8·2-s + 40·4-s + 160·8-s + 2·9-s + 24·11-s + 560·16-s + 16·18-s + 192·22-s − 72·23-s − 10·25-s − 104·29-s + 1.79e3·32-s + 80·36-s + 4·37-s + 960·44-s − 576·46-s − 80·50-s + 92·53-s − 832·58-s + 5.37e3·64-s − 312·67-s + 320·72-s + 32·74-s + 96·79-s + 81·81-s + 3.84e3·88-s − 2.88e3·92-s + ⋯ |
| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 2/9·9-s + 2.18·11-s + 35·16-s + 8/9·18-s + 8.72·22-s − 3.13·23-s − 2/5·25-s − 3.58·29-s + 56·32-s + 20/9·36-s + 4/37·37-s + 21.8·44-s − 12.5·46-s − 8/5·50-s + 1.73·53-s − 14.3·58-s + 84·64-s − 4.65·67-s + 40/9·72-s + 0.432·74-s + 1.21·79-s + 81-s + 43.6·88-s − 31.3·92-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(63.59589642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(63.59589642\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 7 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )( 1 + 4 T + 7 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 5 | $C_2^3$ | \( 1 + 2 p T^{2} - 21 p^{2} T^{4} + 2 p^{5} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 12 T + 169 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 278 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 338 T^{2} + 30723 T^{4} - 338 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 542 T^{2} + 163443 T^{4} + 542 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 36 T + 961 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 1202 T^{2} + 521283 T^{4} + 1202 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T - 1365 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 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 | $C_2^3$ | \( 1 - 2062 T^{2} - 627837 T^{4} - 2062 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 46 T - 693 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 2462 T^{2} - 6055917 T^{4} + 2462 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 7382 T^{2} + 40648083 T^{4} - 7382 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 156 T + 12601 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 674 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 4702 T^{2} - 6289437 T^{4} + 4702 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 48 T + 7009 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4958 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 12002 T^{2} + 81305763 T^{4} - 12002 p^{4} T^{6} + p^{8} T^{8} \) |
| 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
−9.082885522231604893615104091995, −8.314394235003752007890993432851, −8.215943709788261793163736295115, −7.67792022186184128097619617248, −7.49733543773299873491052020990, −7.35576718844514505413826284734, −7.24383162574442680240924842784, −6.70561948652079514970800472245, −6.29371068138964571800578504700, −6.25028326191272454443793978840, −6.09647965588186437682604765896, −5.74132028682906430784234050083, −5.54851762684916186741642815686, −5.14078462655302951999541845045, −4.87244854625212757209097772283, −4.21857986693962474168657053264, −4.13812987247627569108515745904, −3.95495997775410576771610946161, −3.87195546772217061962839448373, −3.30313551901761679008927007053, −3.13292243857505697719644155083, −2.28839238359706004964353527540, −2.08756448051107609927647782481, −1.71734909168165118538751590862, −1.37493838348289676714186710110,
1.37493838348289676714186710110, 1.71734909168165118538751590862, 2.08756448051107609927647782481, 2.28839238359706004964353527540, 3.13292243857505697719644155083, 3.30313551901761679008927007053, 3.87195546772217061962839448373, 3.95495997775410576771610946161, 4.13812987247627569108515745904, 4.21857986693962474168657053264, 4.87244854625212757209097772283, 5.14078462655302951999541845045, 5.54851762684916186741642815686, 5.74132028682906430784234050083, 6.09647965588186437682604765896, 6.25028326191272454443793978840, 6.29371068138964571800578504700, 6.70561948652079514970800472245, 7.24383162574442680240924842784, 7.35576718844514505413826284734, 7.49733543773299873491052020990, 7.67792022186184128097619617248, 8.215943709788261793163736295115, 8.314394235003752007890993432851, 9.082885522231604893615104091995
