| L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s − 4·11-s − 4·13-s + 4·15-s + 8·17-s + 2·19-s − 4·23-s + 3·25-s − 4·27-s + 12·31-s + 8·33-s − 4·37-s + 8·39-s + 8·41-s − 16·43-s − 6·45-s + 4·47-s − 12·49-s − 16·51-s + 16·53-s + 8·55-s − 4·57-s − 8·59-s − 8·61-s + 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s − 1.20·11-s − 1.10·13-s + 1.03·15-s + 1.94·17-s + 0.458·19-s − 0.834·23-s + 3/5·25-s − 0.769·27-s + 2.15·31-s + 1.39·33-s − 0.657·37-s + 1.28·39-s + 1.24·41-s − 2.43·43-s − 0.894·45-s + 0.583·47-s − 1.71·49-s − 2.24·51-s + 2.19·53-s + 1.07·55-s − 0.529·57-s − 1.04·59-s − 1.02·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 80 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 132 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 28 T + 372 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
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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
−8.000044225901767224227961642725, −7.85196029286568146826616631506, −7.35874626088897185190473797697, −7.17848085074397315353680118348, −6.59049727721469305012632514707, −6.50425799242934350531402742359, −5.70646453267605422761103764100, −5.58688743023813479313685275737, −5.18596955333638948176176180447, −4.97975539405571724413945817985, −4.30736205245800156572392018995, −4.28789627647524852567730305147, −3.51832248531185203347291894514, −3.18848012820764256904112270407, −2.68154652196298273452295893037, −2.31678414121467265869669339776, −1.30286531081669497257238006184, −1.12137629004461065882189230491, 0, 0,
1.12137629004461065882189230491, 1.30286531081669497257238006184, 2.31678414121467265869669339776, 2.68154652196298273452295893037, 3.18848012820764256904112270407, 3.51832248531185203347291894514, 4.28789627647524852567730305147, 4.30736205245800156572392018995, 4.97975539405571724413945817985, 5.18596955333638948176176180447, 5.58688743023813479313685275737, 5.70646453267605422761103764100, 6.50425799242934350531402742359, 6.59049727721469305012632514707, 7.17848085074397315353680118348, 7.35874626088897185190473797697, 7.85196029286568146826616631506, 8.000044225901767224227961642725
