| L(s) = 1 | − 4·2-s + 12·4-s − 14·7-s − 32·8-s + 56·14-s + 80·16-s + 20·23-s + 22·25-s − 168·28-s − 192·32-s − 80·46-s + 147·49-s − 88·50-s + 448·56-s + 448·64-s + 220·71-s + 260·79-s + 240·92-s − 588·98-s + 264·100-s − 1.12e3·112-s + 52·113-s + 242·121-s + 127-s − 1.02e3·128-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 4·14-s + 5·16-s + 0.869·23-s + 0.879·25-s − 6·28-s − 6·32-s − 1.73·46-s + 3·49-s − 1.75·50-s + 8·56-s + 7·64-s + 3.09·71-s + 3.29·79-s + 2.60·92-s − 6·98-s + 2.63·100-s − 10·112-s + 0.460·113-s + 2·121-s + 0.00787·127-s − 8·128-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5386008784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5386008784\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 22 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 310 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 650 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1130 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7370 T^{2} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 13130 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−10.61529969008596786945172767000, −10.51087899310759388920598771262, −9.734712996321393937394586878750, −9.727444500556913770571950275583, −9.087430240630928323554829373789, −8.990778939854999818502326918266, −8.363989433624051447360081968953, −7.83860699972286147251623208770, −7.37034042911590247157364555174, −6.82233500543921681153833892962, −6.55994199093782113570087229004, −6.26907253170495291739897754207, −5.57734898536281107431595371262, −4.99442591392303789072079544703, −3.61138370939009327793458227187, −3.49924203571075023747269843157, −2.65238704546857288476558693535, −2.30333505192691213855410668078, −1.09670701450699185155498543735, −0.47139597416106943833822599239,
0.47139597416106943833822599239, 1.09670701450699185155498543735, 2.30333505192691213855410668078, 2.65238704546857288476558693535, 3.49924203571075023747269843157, 3.61138370939009327793458227187, 4.99442591392303789072079544703, 5.57734898536281107431595371262, 6.26907253170495291739897754207, 6.55994199093782113570087229004, 6.82233500543921681153833892962, 7.37034042911590247157364555174, 7.83860699972286147251623208770, 8.363989433624051447360081968953, 8.990778939854999818502326918266, 9.087430240630928323554829373789, 9.727444500556913770571950275583, 9.734712996321393937394586878750, 10.51087899310759388920598771262, 10.61529969008596786945172767000
