| L(s) = 1 | − 6·5-s − 16·13-s + 2·17-s + 19·25-s + 16·29-s + 10·37-s − 32·41-s + 14·49-s + 14·53-s + 10·61-s + 96·65-s + 18·73-s − 12·85-s − 18·89-s − 32·97-s − 30·101-s − 2·109-s + 15·121-s − 66·125-s + 127-s + 131-s + 137-s + 139-s − 96·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 4.43·13-s + 0.485·17-s + 19/5·25-s + 2.97·29-s + 1.64·37-s − 4.99·41-s + 2·49-s + 1.92·53-s + 1.28·61-s + 11.9·65-s + 2.10·73-s − 1.30·85-s − 1.90·89-s − 3.24·97-s − 2.98·101-s − 0.191·109-s + 1.36·121-s − 5.90·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.97·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8713350557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8713350557\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 15 T^{2} + 104 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 25 T^{2} + 264 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 39 T^{2} + 992 T^{4} - 39 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 55 T^{2} + 2064 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 87 T^{2} + 5360 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 7 T - 4 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 111 T^{2} + 8840 T^{4} - 111 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 127 T^{2} + 11640 T^{4} - 127 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−6.59126505120635934808026150108, −6.58938785579002202947655824047, −6.02336448158187654924155466510, −5.95108204420574895342606301619, −5.44571403775347421711467932631, −5.13946341350393984729852944803, −5.11830150325297986493532674103, −5.07376860927659775474446478432, −4.93534479187327599463394063256, −4.47701646489852219843341050654, −4.38412475831685560382062743471, −4.07475957722014555136878686291, −3.96186882139899164203014525417, −3.79976341094494879058115148836, −3.45311620282571182134148518719, −3.02439394334540451085005121174, −2.83927779993662222109080614014, −2.63986301812814163669629027071, −2.62785971823998841128579185016, −2.33063206810995252516373757938, −1.70485322844858430998255331844, −1.53350288105084016813867059960, −0.828907793163875102638107506399, −0.46202895887234430552831230870, −0.35929399108056446612544159993,
0.35929399108056446612544159993, 0.46202895887234430552831230870, 0.828907793163875102638107506399, 1.53350288105084016813867059960, 1.70485322844858430998255331844, 2.33063206810995252516373757938, 2.62785971823998841128579185016, 2.63986301812814163669629027071, 2.83927779993662222109080614014, 3.02439394334540451085005121174, 3.45311620282571182134148518719, 3.79976341094494879058115148836, 3.96186882139899164203014525417, 4.07475957722014555136878686291, 4.38412475831685560382062743471, 4.47701646489852219843341050654, 4.93534479187327599463394063256, 5.07376860927659775474446478432, 5.11830150325297986493532674103, 5.13946341350393984729852944803, 5.44571403775347421711467932631, 5.95108204420574895342606301619, 6.02336448158187654924155466510, 6.58938785579002202947655824047, 6.59126505120635934808026150108
