| L(s) = 1 | + 16·2-s − 54·3-s + 192·4-s + 250·5-s − 864·6-s − 686·7-s + 2.04e3·8-s + 2.18e3·9-s + 4.00e3·10-s + 2.73e3·11-s − 1.03e4·12-s − 8.09e3·13-s − 1.09e4·14-s − 1.35e4·15-s + 2.04e4·16-s + 2.32e4·17-s + 3.49e4·18-s + 2.71e4·19-s + 4.80e4·20-s + 3.70e4·21-s + 4.37e4·22-s − 5.84e4·23-s − 1.10e5·24-s + 4.68e4·25-s − 1.29e5·26-s − 7.87e4·27-s − 1.31e5·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.619·11-s − 1.73·12-s − 1.02·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.14·17-s + 1.41·18-s + 0.907·19-s + 1.34·20-s + 0.872·21-s + 0.876·22-s − 1.00·23-s − 1.63·24-s + 3/5·25-s − 1.44·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.884721496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.884721496\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 2736 T + 6486342 T^{2} - 2736 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8092 T + 64558446 T^{2} + 8092 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 23284 T + 947623654 T^{2} - 23284 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 27136 T + 1834396406 T^{2} - 27136 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 58464 T + 3120126894 T^{2} + 58464 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 17844 T + 27355285806 T^{2} + 17844 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 156376 T + 1963073582 T^{2} - 156376 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 224300 T + 197072716766 T^{2} - 224300 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 649348 T + 439946675638 T^{2} - 649348 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 813400 T + 644025492150 T^{2} - 813400 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 751760 T + 1040931169726 T^{2} + 751760 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 339916 T + 1605220961438 T^{2} + 339916 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1398088 T + 3161100962230 T^{2} - 1398088 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3603836 T + 9516511507022 T^{2} - 3603836 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 528024 T + 12173731088390 T^{2} - 528024 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8153960 T + 34572443743198 T^{2} - 8153960 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10691020 T + 50555673352694 T^{2} - 10691020 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2146832 T + 27921678346398 T^{2} - 2146832 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10477208 T + 76190044109446 T^{2} - 10477208 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3219172 T - 20271096877546 T^{2} - 3219172 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 13140748 T + 165465034912326 T^{2} - 13140748 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.50085666080657913095193482168, −11.05202940986325883018791124247, −10.43363466065597401608564188361, −9.933126220584212312232230035695, −9.613130915863547397955568735790, −9.294308961873056953892403694666, −7.86646309364528630119175976701, −7.79689272896510640670852859668, −6.74234698874957009609034416553, −6.69532082904242932869689671634, −5.93466606209576363244220498841, −5.76955423833586452546709517516, −4.99836507523156849098920509176, −4.82724493498185414808212052690, −3.71981863856695245747671167205, −3.60490982935286719236026726179, −2.34856334242524091663969789996, −2.27075020545133558268794398210, −0.886040292557743200353979105996, −0.831450991981856431241744693296,
0.831450991981856431241744693296, 0.886040292557743200353979105996, 2.27075020545133558268794398210, 2.34856334242524091663969789996, 3.60490982935286719236026726179, 3.71981863856695245747671167205, 4.82724493498185414808212052690, 4.99836507523156849098920509176, 5.76955423833586452546709517516, 5.93466606209576363244220498841, 6.69532082904242932869689671634, 6.74234698874957009609034416553, 7.79689272896510640670852859668, 7.86646309364528630119175976701, 9.294308961873056953892403694666, 9.613130915863547397955568735790, 9.933126220584212312232230035695, 10.43363466065597401608564188361, 11.05202940986325883018791124247, 11.50085666080657913095193482168
