| L(s) = 1 | − 128·4-s − 1.50e3·7-s + 1.63e4·16-s − 1.54e5·25-s + 1.93e5·28-s + 6.62e5·31-s + 5.84e4·49-s − 2.09e6·64-s − 5.63e6·73-s − 1.51e7·79-s − 2.34e7·97-s + 1.97e7·100-s + 3.62e7·103-s − 2.47e7·112-s − 6.26e6·121-s − 8.48e7·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.25e8·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 1.66·7-s + 16-s − 1.97·25-s + 1.66·28-s + 3.99·31-s + 0.0709·49-s − 64-s − 1.69·73-s − 3.45·79-s − 2.61·97-s + 1.97·100-s + 3.26·103-s − 1.66·112-s − 0.321·121-s − 3.99·124-s + 2·169-s + 3.28·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5236968080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5236968080\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 154222 T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 754 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 6264730 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 28883912350 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 331370 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2292036335474 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2785421000870 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2819362 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7561270 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 15670457974954 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 11745214 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.24863772841221404555557794180, −13.30011263114948818579681356481, −12.46134326539530540001033827274, −11.87209134335341888149725856594, −11.45932674267913354402041858448, −10.20116296391575939678359287549, −9.998866218785554060420595236770, −9.742452573279157819528930403602, −8.925775764603181529601169411735, −8.341158301292692795896627008245, −7.77588969726576338970570623994, −6.86620070201066839629689654187, −6.14707998115043803754847396476, −5.81567048771579104032684665277, −4.66960483733547722619271184263, −4.16012687790846032521296712357, −3.27455762753789063951133729206, −2.68888608667737095455828658446, −1.25733569173564249618206380507, −0.27817727734191117039697205862,
0.27817727734191117039697205862, 1.25733569173564249618206380507, 2.68888608667737095455828658446, 3.27455762753789063951133729206, 4.16012687790846032521296712357, 4.66960483733547722619271184263, 5.81567048771579104032684665277, 6.14707998115043803754847396476, 6.86620070201066839629689654187, 7.77588969726576338970570623994, 8.341158301292692795896627008245, 8.925775764603181529601169411735, 9.742452573279157819528930403602, 9.998866218785554060420595236770, 10.20116296391575939678359287549, 11.45932674267913354402041858448, 11.87209134335341888149725856594, 12.46134326539530540001033827274, 13.30011263114948818579681356481, 14.24863772841221404555557794180
