| L(s) = 1 | + 2·3-s + 2·4-s − 8·9-s + 4·12-s + 36·13-s − 16-s + 12·19-s − 22·27-s + 136·31-s − 16·36-s − 16·37-s + 72·39-s + 160·43-s − 2·48-s + 14·49-s + 72·52-s + 24·57-s − 156·61-s + 20·64-s + 24·67-s + 32·73-s + 24·76-s + 128·79-s + 7·81-s + 272·93-s + 8·97-s + 352·103-s + ⋯ |
| L(s) = 1 | + 2/3·3-s + 1/2·4-s − 8/9·9-s + 1/3·12-s + 2.76·13-s − 0.0625·16-s + 0.631·19-s − 0.814·27-s + 4.38·31-s − 4/9·36-s − 0.432·37-s + 1.84·39-s + 3.72·43-s − 0.0416·48-s + 2/7·49-s + 1.38·52-s + 8/19·57-s − 2.55·61-s + 5/16·64-s + 0.358·67-s + 0.438·73-s + 6/19·76-s + 1.62·79-s + 7/81·81-s + 2.92·93-s + 8/97·97-s + 3.41·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.59320082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.59320082\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $D_{4}$ | \( 1 - 2 T + 4 p T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - p T^{2} + 5 T^{4} - p^{5} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 428 T^{2} + 74630 T^{4} - 428 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 18 T + 412 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 988 T^{2} + 407046 T^{4} - 988 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 556 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1444 T^{2} + 980166 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 2972 T^{2} + 3611558 T^{4} - 2972 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 68 T + 3050 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 1382 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 1292 T^{2} + 2832038 T^{4} - 1292 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 80 T + 5046 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 6148 T^{2} + 19144326 T^{4} - 6148 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} - 13350138 T^{4} + 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 3676 T^{2} + 15964266 T^{4} - 3676 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 78 T + 8620 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 8566 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 10588 T^{2} + 78813510 T^{4} - 10588 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 16 T + 5990 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 64 T + 4434 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 13948 T^{2} + 141899946 T^{4} - 13948 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 11468 T^{2} + 120945830 T^{4} - 11468 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 18374 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.60121438920780280796724678348, −7.47799367831022913100146923709, −7.30495068543521718695131673484, −6.82065493903040063181416200135, −6.47286088224640750201629114869, −6.36114039056143446881667655754, −6.19141050016745642068693710342, −5.92479347971712120840053558528, −5.91683204446669715893319979958, −5.43748925662598567356925453385, −5.23032445799730882626244076864, −4.76333628650102441789081302355, −4.35207298355642828610298411571, −4.26489584720041490425551361106, −4.24120229983762025144076543010, −3.50467545047967323818612310822, −3.19378917865023948813616325932, −3.13849589369157522147513808300, −3.06254019104411869433394972725, −2.39305564294611814956463855922, −2.14199195310634910889420714450, −1.90702500622157807192884417083, −0.999771300757133272744418340507, −0.963362686823475302955792537051, −0.70147442027864791840994660208,
0.70147442027864791840994660208, 0.963362686823475302955792537051, 0.999771300757133272744418340507, 1.90702500622157807192884417083, 2.14199195310634910889420714450, 2.39305564294611814956463855922, 3.06254019104411869433394972725, 3.13849589369157522147513808300, 3.19378917865023948813616325932, 3.50467545047967323818612310822, 4.24120229983762025144076543010, 4.26489584720041490425551361106, 4.35207298355642828610298411571, 4.76333628650102441789081302355, 5.23032445799730882626244076864, 5.43748925662598567356925453385, 5.91683204446669715893319979958, 5.92479347971712120840053558528, 6.19141050016745642068693710342, 6.36114039056143446881667655754, 6.47286088224640750201629114869, 6.82065493903040063181416200135, 7.30495068543521718695131673484, 7.47799367831022913100146923709, 7.60121438920780280796724678348
