| L(s) = 1 | + 2·2-s + 3·4-s + 4·5-s + 4·8-s + 8·10-s − 2·11-s + 6·13-s + 5·16-s + 14·17-s + 2·19-s + 12·20-s − 4·22-s − 2·23-s + 5·25-s + 12·26-s − 10·29-s − 6·31-s + 6·32-s + 28·34-s + 4·37-s + 4·38-s + 16·40-s + 12·41-s + 4·43-s − 6·44-s − 4·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.78·5-s + 1.41·8-s + 2.52·10-s − 0.603·11-s + 1.66·13-s + 5/4·16-s + 3.39·17-s + 0.458·19-s + 2.68·20-s − 0.852·22-s − 0.417·23-s + 25-s + 2.35·26-s − 1.85·29-s − 1.07·31-s + 1.06·32-s + 4.80·34-s + 0.657·37-s + 0.648·38-s + 2.52·40-s + 1.87·41-s + 0.609·43-s − 0.904·44-s − 0.589·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.48000772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.48000772\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 230 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 243 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 76 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.63154737740895890823531401902, −7.57712240718078775141091545634, −7.35062694648888552997509807419, −7.06247567973928688257141438554, −6.06279404806978818411957296295, −6.05193986313449556641568717912, −5.84296773517598862317108854069, −5.77623355945256934844047712964, −5.29895056631067527230210061807, −5.21927104765924035335365372707, −4.25793106883908542110472643225, −4.24528162004751063292812752860, −3.55721559993881827530076958212, −3.47672204224157539676236419226, −2.88682126596248651545546716194, −2.68448121379884614987553028818, −1.91446519504405065760906291454, −1.78497048644836576799763715775, −1.19089669879646583597625937337, −0.846087250399229237515631728529,
0.846087250399229237515631728529, 1.19089669879646583597625937337, 1.78497048644836576799763715775, 1.91446519504405065760906291454, 2.68448121379884614987553028818, 2.88682126596248651545546716194, 3.47672204224157539676236419226, 3.55721559993881827530076958212, 4.24528162004751063292812752860, 4.25793106883908542110472643225, 5.21927104765924035335365372707, 5.29895056631067527230210061807, 5.77623355945256934844047712964, 5.84296773517598862317108854069, 6.05193986313449556641568717912, 6.06279404806978818411957296295, 7.06247567973928688257141438554, 7.35062694648888552997509807419, 7.57712240718078775141091545634, 7.63154737740895890823531401902
