| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 4·11-s + 35·16-s + 16·22-s + 8·23-s − 4·25-s + 16·31-s + 56·32-s + 8·37-s + 8·43-s + 40·44-s + 32·46-s − 16·47-s − 16·50-s − 8·53-s − 32·59-s + 16·61-s + 64·62-s + 84·64-s + 16·67-s + 32·74-s + 16·79-s + 32·86-s + 80·88-s − 16·89-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 1.20·11-s + 35/4·16-s + 3.41·22-s + 1.66·23-s − 4/5·25-s + 2.87·31-s + 9.89·32-s + 1.31·37-s + 1.21·43-s + 6.03·44-s + 4.71·46-s − 2.33·47-s − 2.26·50-s − 1.09·53-s − 4.16·59-s + 2.04·61-s + 8.12·62-s + 21/2·64-s + 1.95·67-s + 3.71·74-s + 1.80·79-s + 3.45·86-s + 8.52·88-s − 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(109.6123430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(109.6123430\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 4 T^{2} + 46 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T^{2} - 32 T^{3} + 242 T^{4} - 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 36 T^{2} - 32 T^{3} + 638 T^{4} - 32 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 40 T^{2} + 80 T^{3} + 794 T^{4} + 80 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 36 T^{2} + 56 T^{3} - 570 T^{4} + 56 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 200 T^{2} - 1568 T^{3} + 10378 T^{4} - 1568 p T^{5} + 200 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 76 T^{2} - 408 T^{3} + 2486 T^{4} - 408 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 132 T^{2} + 32 T^{3} + 7454 T^{4} + 32 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 116 T^{2} - 424 T^{3} + 5110 T^{4} - 424 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 248 T^{2} + 2304 T^{3} + 18890 T^{4} + 2304 p T^{5} + 248 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 140 T^{2} + 1048 T^{3} + 9334 T^{4} + 1048 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 304 T^{2} - 2864 T^{3} + 29362 T^{4} - 2864 p T^{5} + 304 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 124 T^{2} - 1104 T^{3} + 11510 T^{4} - 1104 p T^{5} + 124 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 140 T^{2} + 256 T^{3} + 12422 T^{4} + 256 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 116 T^{2} - 448 T^{3} + 6462 T^{4} - 448 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 236 T^{2} - 2448 T^{3} + 24582 T^{4} - 2448 p T^{5} + 236 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 280 T^{2} + 48 T^{3} + 32794 T^{4} + 48 p T^{5} + 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 160 T^{2} + 1904 T^{3} + 24642 T^{4} + 1904 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 352 T^{2} + 3984 T^{3} + 51458 T^{4} + 3984 p T^{5} + 352 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.48552228788439949160136406502, −5.24148545148005391033831629515, −4.94508857249820285055012393814, −4.70234835600549987876508252077, −4.62522315205475673158084260645, −4.45145323258622128395809210767, −4.31045226840597124346438033659, −4.17025833486356583580176413556, −4.04060077385134389081472125588, −3.87970392220113667890541132211, −3.41278598763777026428041312173, −3.33292651936476729224546006944, −3.23325929674411569881889109414, −2.97486351769955746842570513503, −2.80704125945089637808992115212, −2.73387894166343282327866351614, −2.59993280436734521194587103711, −2.00334853676016076889840769628, −1.90018564425422211803507168312, −1.83309377357171726648245547131, −1.56558271460816864309062141549, −1.25051474787283699224361739118, −0.925643581893000526428842730780, −0.59569537409131425080546109122, −0.59201217388398950288287038429,
0.59201217388398950288287038429, 0.59569537409131425080546109122, 0.925643581893000526428842730780, 1.25051474787283699224361739118, 1.56558271460816864309062141549, 1.83309377357171726648245547131, 1.90018564425422211803507168312, 2.00334853676016076889840769628, 2.59993280436734521194587103711, 2.73387894166343282327866351614, 2.80704125945089637808992115212, 2.97486351769955746842570513503, 3.23325929674411569881889109414, 3.33292651936476729224546006944, 3.41278598763777026428041312173, 3.87970392220113667890541132211, 4.04060077385134389081472125588, 4.17025833486356583580176413556, 4.31045226840597124346438033659, 4.45145323258622128395809210767, 4.62522315205475673158084260645, 4.70234835600549987876508252077, 4.94508857249820285055012393814, 5.24148545148005391033831629515, 5.48552228788439949160136406502
