| L(s) = 1 | + 288·11-s − 128·16-s − 3.31e3·71-s + 782·81-s + 4.65e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 3.68e4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 7.89·11-s − 2·16-s − 5.53·71-s + 1.07·81-s + 34.9·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.000439·173-s − 15.7·176-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.101643981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.101643981\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )^{2}( 1 + p^{2} T + p^{3} T^{2} )^{2} \) |
| 3 | $C_2^3$ | \( 1 - 782 T^{4} + p^{12} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 1958542 T^{4} + p^{12} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 47706622 T^{4} + p^{12} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 45862 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 9293086658 T^{4} + p^{12} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 828 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 48396356062 T^{4} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 930382 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 648698638898 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 859766289982 T^{4} + p^{12} T^{8} \) |
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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
−8.868532248413253281775039982904, −8.818115110956106691306981512147, −8.796926767023848804468996843658, −8.133286232909568918938611978457, −7.47585598746891266061788930813, −7.24925031778856953017875058448, −7.14853088590254057043267955985, −6.71044401648984404762185379160, −6.51132032827171614275763535128, −6.42198528152021937092225601303, −6.18789814020095095963493605097, −6.03589896687418892999289076625, −5.45477762558318946880396008311, −4.79043776192912662876585867725, −4.39785724604576471865363608145, −4.18709959297759714545216324726, −4.16718971666715934557717030565, −3.89292955722487230013523874397, −3.29703462020743613994706841993, −3.24000817839465725477859038714, −2.24568408063567922627201482956, −1.68076497642668533762486257364, −1.39536160318080882037720926485, −1.30682075622832443667470256639, −0.59563017072643370177712322020,
0.59563017072643370177712322020, 1.30682075622832443667470256639, 1.39536160318080882037720926485, 1.68076497642668533762486257364, 2.24568408063567922627201482956, 3.24000817839465725477859038714, 3.29703462020743613994706841993, 3.89292955722487230013523874397, 4.16718971666715934557717030565, 4.18709959297759714545216324726, 4.39785724604576471865363608145, 4.79043776192912662876585867725, 5.45477762558318946880396008311, 6.03589896687418892999289076625, 6.18789814020095095963493605097, 6.42198528152021937092225601303, 6.51132032827171614275763535128, 6.71044401648984404762185379160, 7.14853088590254057043267955985, 7.24925031778856953017875058448, 7.47585598746891266061788930813, 8.133286232909568918938611978457, 8.796926767023848804468996843658, 8.818115110956106691306981512147, 8.868532248413253281775039982904
