| L(s) = 1 | + 4·3-s + 2·9-s − 4·11-s + 16·19-s − 6·25-s − 16·27-s − 16·33-s + 8·41-s + 8·43-s − 8·49-s + 64·57-s + 28·59-s + 36·67-s − 8·73-s − 24·75-s − 31·81-s − 8·83-s − 44·89-s + 28·97-s − 8·99-s − 8·107-s − 44·113-s + 10·121-s + 32·123-s + 127-s + 32·129-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2/3·9-s − 1.20·11-s + 3.67·19-s − 6/5·25-s − 3.07·27-s − 2.78·33-s + 1.24·41-s + 1.21·43-s − 8/7·49-s + 8.47·57-s + 3.64·59-s + 4.39·67-s − 0.936·73-s − 2.77·75-s − 3.44·81-s − 0.878·83-s − 4.66·89-s + 2.84·97-s − 0.804·99-s − 0.773·107-s − 4.13·113-s + 0.909·121-s + 2.88·123-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.180541983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.180541983\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_{4}$ | \( ( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 6 T^{2} + 27 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 8 T^{2} + 82 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 32 T^{2} + 562 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 82 T^{2} + 2731 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 4 T^{2} - 362 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 114 T^{2} + 5163 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 42 T^{2} + 3147 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 60 T^{2} + 3270 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 20 T^{2} + 5590 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 14 T + 149 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 84 T^{2} + 7158 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 18 T + 213 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 114 T^{2} + 11019 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 100 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 276 T^{2} + 31398 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 22 T + 291 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
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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
−6.20171764183651069716213277131, −5.77813802848659127103954180334, −5.73155483662435654064715359858, −5.55913870487984290720916971314, −5.54173611252307677551332970614, −5.19983034541674288678741982944, −5.08340713434175382779750490007, −5.01937980748798547126661224347, −4.42849636562157339606679111411, −4.17801401775910361106475225571, −3.98810730401219364994225612858, −3.79180657489314491980959355205, −3.77531874840887366866188853898, −3.23334292758422332800542072948, −3.01691843571609902426634997713, −2.97031150616081852825930030042, −2.92344694522445615040402391391, −2.67931160681426183502726149282, −2.30133515583992129114583295039, −2.03341130218715305518467199579, −2.00468675312629959696449544872, −1.50528443120717451299076346950, −0.989928506464978586324354401541, −0.76294282386679652378221676559, −0.41377550735793894382686408593,
0.41377550735793894382686408593, 0.76294282386679652378221676559, 0.989928506464978586324354401541, 1.50528443120717451299076346950, 2.00468675312629959696449544872, 2.03341130218715305518467199579, 2.30133515583992129114583295039, 2.67931160681426183502726149282, 2.92344694522445615040402391391, 2.97031150616081852825930030042, 3.01691843571609902426634997713, 3.23334292758422332800542072948, 3.77531874840887366866188853898, 3.79180657489314491980959355205, 3.98810730401219364994225612858, 4.17801401775910361106475225571, 4.42849636562157339606679111411, 5.01937980748798547126661224347, 5.08340713434175382779750490007, 5.19983034541674288678741982944, 5.54173611252307677551332970614, 5.55913870487984290720916971314, 5.73155483662435654064715359858, 5.77813802848659127103954180334, 6.20171764183651069716213277131
