| L(s) = 1 | − 2·2-s + 4-s − 2·8-s + 9-s + 2·11-s − 2·18-s − 4·22-s + 12·23-s − 7·25-s − 2·29-s + 14·32-s + 36-s − 20·37-s − 14·43-s + 2·44-s − 24·46-s + 14·50-s + 24·53-s + 4·58-s − 17·64-s − 4·67-s − 16·71-s − 2·72-s + 40·74-s − 12·79-s − 14·81-s + 28·86-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.707·8-s + 1/3·9-s + 0.603·11-s − 0.471·18-s − 0.852·22-s + 2.50·23-s − 7/5·25-s − 0.371·29-s + 2.47·32-s + 1/6·36-s − 3.28·37-s − 2.13·43-s + 0.301·44-s − 3.53·46-s + 1.97·50-s + 3.29·53-s + 0.525·58-s − 2.12·64-s − 0.488·67-s − 1.89·71-s − 0.235·72-s + 4.64·74-s − 1.35·79-s − 1.55·81-s + 3.01·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3029211820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3029211820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( ( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^2:C_4$ | \( 1 - T^{2} + 5 p T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 7 T^{2} + 59 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - T - 7 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 5 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 63 T^{2} + 1711 T^{4} + 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 55 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 72 T^{2} + 2581 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 86 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 8 T^{2} + 2910 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 17 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 32 T^{2} - 1059 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 129 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 184 T^{2} + 15101 T^{4} + 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 192 T^{2} + 16606 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 240 T^{2} + 25006 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 280 T^{2} + 33053 T^{4} + 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 239 T^{2} + 29859 T^{4} + 239 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 141 T^{2} + 19807 T^{4} + 141 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.58441032063065844183125742051, −5.30550466669453878398302411373, −5.18232574764156910661665875176, −5.00672996832510424290660335529, −4.63648229161852666449257432764, −4.51739259526234953127038261307, −4.38785186646930471192366119893, −4.25290009521068291650167707685, −3.94071070549612906294765072153, −3.81367520020455237225219239822, −3.54147355368751572611840103133, −3.16931935388925376355283627412, −3.14054159572826942368238872582, −3.11140668115273357675203638728, −2.81026517375331879482562573813, −2.56088851577980221179912887519, −2.11525785735425788266658815024, −2.09441455474217205252611338083, −1.68770265140572046663185244764, −1.62356816974296849758901703186, −1.50099855986213935497839866693, −0.862613518979998293977666889088, −0.823601577912411213656678380839, −0.49531856070579722067489618239, −0.12367880362586509526980546787,
0.12367880362586509526980546787, 0.49531856070579722067489618239, 0.823601577912411213656678380839, 0.862613518979998293977666889088, 1.50099855986213935497839866693, 1.62356816974296849758901703186, 1.68770265140572046663185244764, 2.09441455474217205252611338083, 2.11525785735425788266658815024, 2.56088851577980221179912887519, 2.81026517375331879482562573813, 3.11140668115273357675203638728, 3.14054159572826942368238872582, 3.16931935388925376355283627412, 3.54147355368751572611840103133, 3.81367520020455237225219239822, 3.94071070549612906294765072153, 4.25290009521068291650167707685, 4.38785186646930471192366119893, 4.51739259526234953127038261307, 4.63648229161852666449257432764, 5.00672996832510424290660335529, 5.18232574764156910661665875176, 5.30550466669453878398302411373, 5.58441032063065844183125742051
