L(s) = 1 | − 2·4-s − 2·9-s + 10·11-s + 3·16-s − 4·19-s − 16·29-s + 22·31-s + 4·36-s − 4·41-s − 20·44-s + 21·49-s + 10·59-s + 20·61-s − 4·64-s + 40·71-s + 8·76-s + 18·79-s + 3·81-s − 32·89-s − 20·99-s + 22·101-s − 60·109-s + 32·116-s + 21·121-s − 44·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 4-s − 2/3·9-s + 3.01·11-s + 3/4·16-s − 0.917·19-s − 2.97·29-s + 3.95·31-s + 2/3·36-s − 0.624·41-s − 3.01·44-s + 3·49-s + 1.30·59-s + 2.56·61-s − 1/2·64-s + 4.74·71-s + 0.917·76-s + 2.02·79-s + 1/3·81-s − 3.39·89-s − 2.01·99-s + 2.18·101-s − 5.74·109-s + 2.97·116-s + 1.90·121-s − 3.95·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.747986068\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.747986068\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 197 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 22 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1342 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_4$ | \( ( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4$ | \( ( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 5118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 494 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 7993 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 9382 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 9 T + 177 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 27917 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 345 T^{2} + 48473 T^{4} - 345 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.08978075252129906517765558394, −5.70357608539399250199509653455, −5.56167598858894034674322289281, −5.39931963248966999601693959755, −5.25022517195596392602221165246, −5.10394689678368055648557169948, −4.91596546278039138270145106310, −4.41420496348118031964935637685, −4.32018699830566496315850034025, −4.05212000828020489745152026707, −3.97302534598986944478945281617, −3.91073738601513652425396170018, −3.71141532505627431820454143280, −3.51872437844917882741318008169, −3.24250830238277051580829167914, −2.71779136801374832016358090500, −2.58567629558690599962904283517, −2.48181653355048834961349957212, −2.03590375370981372238896888714, −2.02143883199781535915446479876, −1.37937092007974387497095449482, −1.23971743193916980544054294320, −1.01253104172303924867648600466, −0.61612158852901280846381102638, −0.41129575226261045029737384163,
0.41129575226261045029737384163, 0.61612158852901280846381102638, 1.01253104172303924867648600466, 1.23971743193916980544054294320, 1.37937092007974387497095449482, 2.02143883199781535915446479876, 2.03590375370981372238896888714, 2.48181653355048834961349957212, 2.58567629558690599962904283517, 2.71779136801374832016358090500, 3.24250830238277051580829167914, 3.51872437844917882741318008169, 3.71141532505627431820454143280, 3.91073738601513652425396170018, 3.97302534598986944478945281617, 4.05212000828020489745152026707, 4.32018699830566496315850034025, 4.41420496348118031964935637685, 4.91596546278039138270145106310, 5.10394689678368055648557169948, 5.25022517195596392602221165246, 5.39931963248966999601693959755, 5.56167598858894034674322289281, 5.70357608539399250199509653455, 6.08978075252129906517765558394