| L(s) = 1 | + 44·7-s + 388·25-s − 736·37-s + 760·43-s + 766·49-s − 1.18e3·67-s − 3.34e3·79-s − 3.37e3·109-s + 4.84e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.01e3·169-s + 173-s + 1.70e4·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2.37·7-s + 3.10·25-s − 3.27·37-s + 2.69·43-s + 2.23·49-s − 2.15·67-s − 4.76·79-s − 2.96·109-s + 3.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.19·169-s + 0.000439·173-s + 7.37·175-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.125298465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125298465\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3506 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 5830 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 5726 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 18284 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 32936 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 2750 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 184 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 126742 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 190 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 205870 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 169736 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 403654 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 24170 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 296 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 607244 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130682 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 836 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 350042 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 926422 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1504778 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−6.84240309975801776018506888278, −6.76298995575358418315956786084, −6.20605503194597199735631211975, −6.02335393841764142501071162557, −5.60226913390946430457912466678, −5.54604821581197858399795919200, −5.49830322301098974848701494566, −5.07611050077463133043574571028, −4.81770872529146346674714293746, −4.58581311774775998305184631169, −4.57289845668700493157914744029, −4.32032931461601262676630185666, −3.90237207799357689496829756303, −3.79775957083581739833083276745, −3.23348709342511357400160073399, −3.01264545073106386118664557784, −2.82147345830056843226560733987, −2.67384943902261968679828232098, −2.03081816131703258115708863376, −1.92346557314408369211789909655, −1.52850324920843157759221719503, −1.36011212350590107593851550640, −1.06625760258022828552797566299, −0.71583547953780455900946877674, −0.18062066221630131497171129441,
0.18062066221630131497171129441, 0.71583547953780455900946877674, 1.06625760258022828552797566299, 1.36011212350590107593851550640, 1.52850324920843157759221719503, 1.92346557314408369211789909655, 2.03081816131703258115708863376, 2.67384943902261968679828232098, 2.82147345830056843226560733987, 3.01264545073106386118664557784, 3.23348709342511357400160073399, 3.79775957083581739833083276745, 3.90237207799357689496829756303, 4.32032931461601262676630185666, 4.57289845668700493157914744029, 4.58581311774775998305184631169, 4.81770872529146346674714293746, 5.07611050077463133043574571028, 5.49830322301098974848701494566, 5.54604821581197858399795919200, 5.60226913390946430457912466678, 6.02335393841764142501071162557, 6.20605503194597199735631211975, 6.76298995575358418315956786084, 6.84240309975801776018506888278
