| L(s) = 1 | − 2·2-s + 4-s − 6·5-s + 2·8-s + 12·10-s − 6·11-s − 4·13-s − 4·16-s − 3·17-s − 10·19-s − 6·20-s + 12·22-s + 18·23-s + 19·25-s + 8·26-s − 6·29-s − 4·31-s + 2·32-s + 6·34-s − 4·37-s + 20·38-s − 12·40-s + 15·41-s − 43-s − 6·44-s − 36·46-s − 38·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 2.68·5-s + 0.707·8-s + 3.79·10-s − 1.80·11-s − 1.10·13-s − 16-s − 0.727·17-s − 2.29·19-s − 1.34·20-s + 2.55·22-s + 3.75·23-s + 19/5·25-s + 1.56·26-s − 1.11·29-s − 0.718·31-s + 0.353·32-s + 1.02·34-s − 0.657·37-s + 3.24·38-s − 1.89·40-s + 2.34·41-s − 0.152·43-s − 0.904·44-s − 5.30·46-s − 5.37·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06610189303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06610189303\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 6 T - 46 T^{2} - 144 T^{3} + 2007 T^{4} - 144 p T^{5} - 46 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 216 p T^{5} - 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 484 p T^{5} + 43 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 7 T - 35 T^{2} - 434 T^{3} - 1850 T^{4} - 434 p T^{5} - 35 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 1152 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 864 p T^{5} + 98 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 74 p T^{5} - 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.51644988432927878738453922655, −6.34596704540668470434794396383, −5.83893999500945276131969018199, −5.52962542914639412817328140813, −5.23127497932271184477592250270, −5.11973514008131197406916928648, −5.11020444346490057931708584325, −5.03840790656427580564315476546, −4.34871802716806614813430058164, −4.32830698603519802808455166401, −4.30515652591084021932677459827, −4.03275428007483714887243676968, −3.79336767974122541992016669289, −3.63116879773292896030957052502, −3.07596571046677670691874317417, −2.95396688870024918370384768281, −2.67074769297286086357467852376, −2.55381423426269280673360830379, −2.52317121671990798366730550905, −1.84311554275932517399960778460, −1.55152617679089887053512646638, −1.26947623361411500611525474621, −0.817645971777291304580223704905, −0.25346285820341548265606375768, −0.20493618669316302752020384876,
0.20493618669316302752020384876, 0.25346285820341548265606375768, 0.817645971777291304580223704905, 1.26947623361411500611525474621, 1.55152617679089887053512646638, 1.84311554275932517399960778460, 2.52317121671990798366730550905, 2.55381423426269280673360830379, 2.67074769297286086357467852376, 2.95396688870024918370384768281, 3.07596571046677670691874317417, 3.63116879773292896030957052502, 3.79336767974122541992016669289, 4.03275428007483714887243676968, 4.30515652591084021932677459827, 4.32830698603519802808455166401, 4.34871802716806614813430058164, 5.03840790656427580564315476546, 5.11020444346490057931708584325, 5.11973514008131197406916928648, 5.23127497932271184477592250270, 5.52962542914639412817328140813, 5.83893999500945276131969018199, 6.34596704540668470434794396383, 6.51644988432927878738453922655
