| L(s) = 1 | + 2·2-s + 4-s + 7-s − 2·8-s − 2·9-s − 2·11-s + 2·14-s − 4·16-s − 4·18-s − 4·22-s + 2·23-s − 2·25-s + 28-s − 2·29-s − 2·32-s − 2·36-s + 2·43-s − 2·44-s + 4·46-s − 4·50-s − 2·53-s − 2·56-s − 4·58-s − 2·63-s + 3·64-s − 2·67-s + 4·72-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s + 7-s − 2·8-s − 2·9-s − 2·11-s + 2·14-s − 4·16-s − 4·18-s − 4·22-s + 2·23-s − 2·25-s + 28-s − 2·29-s − 2·32-s − 2·36-s + 2·43-s − 2·44-s + 4·46-s − 4·50-s − 2·53-s − 2·56-s − 4·58-s − 2·63-s + 3·64-s − 2·67-s + 4·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.109297319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109297319\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.068185897651109290202133155642, −8.616393398478971506303617956070, −8.165212333626475066163691916678, −7.82379562498106380316534201331, −7.66046299251959614395475364462, −7.11147937566692854222703472103, −6.43539811039768314637720029380, −5.95656712289839372661099339297, −5.72759633888322940511678949728, −5.45498497542622004492528608558, −5.33677324405041114899652064935, −4.69378516200538305512138604879, −4.63504284814224181809837441960, −4.04142732529596503309845800930, −3.48091883111733804671273564075, −3.05211325529967211730292776187, −2.91906473359937282052904275909, −2.33921327199187931071240965184, −1.87545506818864344996316884521, −0.42002563221972887963389766486,
0.42002563221972887963389766486, 1.87545506818864344996316884521, 2.33921327199187931071240965184, 2.91906473359937282052904275909, 3.05211325529967211730292776187, 3.48091883111733804671273564075, 4.04142732529596503309845800930, 4.63504284814224181809837441960, 4.69378516200538305512138604879, 5.33677324405041114899652064935, 5.45498497542622004492528608558, 5.72759633888322940511678949728, 5.95656712289839372661099339297, 6.43539811039768314637720029380, 7.11147937566692854222703472103, 7.66046299251959614395475364462, 7.82379562498106380316534201331, 8.165212333626475066163691916678, 8.616393398478971506303617956070, 9.068185897651109290202133155642
