| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 7-s − 4·8-s + 3·9-s + 4·10-s + 2·11-s − 6·12-s − 2·13-s + 2·14-s + 4·15-s + 5·16-s − 2·17-s − 6·18-s − 8·19-s − 6·20-s + 2·21-s − 4·22-s + 9·23-s + 8·24-s + 3·25-s + 4·26-s − 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s + 1.26·10-s + 0.603·11-s − 1.73·12-s − 0.554·13-s + 0.534·14-s + 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 1.83·19-s − 1.34·20-s + 0.436·21-s − 0.852·22-s + 1.87·23-s + 1.63·24-s + 3/5·25-s + 0.784·26-s − 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4914157823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4914157823\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T - 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 206 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 176 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.357841920337290323968175477710, −8.026177963500719243738330912203, −7.48742552501311536379806758440, −7.26741365649908379382846829192, −6.88117386658057782635738793468, −6.74993415080249879938369858306, −6.24717543828721058977862492233, −6.14561448837823693786039555317, −5.43452172987535784232171198628, −5.19602631471288819476757639719, −4.61449278280852034215899450891, −4.41134255697790960017042244564, −3.74034202015720985244498576941, −3.61484810425167932274722406702, −2.82530654605725307722853002095, −2.55063128601546482485882709861, −1.73973230800917795382689808226, −1.58097132559433454281580946220, −0.51895233004753210864036658985, −0.50995253548783907239935113684,
0.50995253548783907239935113684, 0.51895233004753210864036658985, 1.58097132559433454281580946220, 1.73973230800917795382689808226, 2.55063128601546482485882709861, 2.82530654605725307722853002095, 3.61484810425167932274722406702, 3.74034202015720985244498576941, 4.41134255697790960017042244564, 4.61449278280852034215899450891, 5.19602631471288819476757639719, 5.43452172987535784232171198628, 6.14561448837823693786039555317, 6.24717543828721058977862492233, 6.74993415080249879938369858306, 6.88117386658057782635738793468, 7.26741365649908379382846829192, 7.48742552501311536379806758440, 8.026177963500719243738330912203, 8.357841920337290323968175477710
