| L(s) = 1 | − 4·2-s − 9·3-s − 24·5-s + 36·6-s + 64·8-s + 96·10-s − 66·11-s + 196·13-s + 216·15-s − 256·16-s + 216·17-s + 340·19-s + 264·22-s + 1.03e3·23-s − 576·24-s + 3.12e3·25-s − 784·26-s + 729·27-s − 4.98e3·29-s − 864·30-s + 7.04e3·31-s + 594·33-s − 864·34-s + 1.22e4·37-s − 1.36e3·38-s − 1.76e3·39-s − 1.53e3·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.429·5-s + 0.408·6-s + 0.353·8-s + 0.303·10-s − 0.164·11-s + 0.321·13-s + 0.247·15-s − 1/4·16-s + 0.181·17-s + 0.216·19-s + 0.116·22-s + 0.409·23-s − 0.204·24-s + 25-s − 0.227·26-s + 0.192·27-s − 1.09·29-s − 0.175·30-s + 1.31·31-s + 0.0949·33-s − 0.128·34-s + 1.46·37-s − 0.152·38-s − 0.185·39-s − 0.151·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.08196561905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08196561905\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 24 T - 2549 T^{2} + 24 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 p T - 1295 p^{2} T^{2} + 6 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 98 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 216 T - 1373201 T^{2} - 216 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 340 T - 2360499 T^{2} - 340 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1038 T - 5358899 T^{2} - 1038 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2490 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 7048 T + 21045153 T^{2} - 7048 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 12238 T + 80424687 T^{2} - 12238 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6468 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 15412 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 20604 T + 195179809 T^{2} + 20604 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 32490 T + 637404607 T^{2} + 32490 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 34224 T + 456357877 T^{2} + 34224 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 35654 T + 426611415 T^{2} + 35654 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12680 T - 1189342707 T^{2} + 12680 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 42642 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 33734 T - 935088837 T^{2} + 33734 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 85108 T + 4166315265 T^{2} - 85108 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 106764 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 34884 T - 4367165993 T^{2} + 34884 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18662 T + p^{5} T^{2} )^{2} \) |
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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
−11.30579771906466616373457624666, −10.69130732031347884104465603253, −10.26816884732463667571960251537, −9.565308963540641386559321365447, −9.536254148590375068907934523476, −8.586118616181049151805917464997, −8.508014197101937328426509615105, −7.66655130267807684623141202925, −7.58307450135194912509226140097, −6.76468939624162810992345830483, −6.27630639084589703059708576624, −5.84865193661912456967108777962, −4.98500003939528906685647104033, −4.70422700344555341511780918719, −4.05343344979342324821249028099, −3.12373145515759498350568064050, −2.80181462649680150735487422175, −1.48987807716648405614549165875, −1.21454759450132784891243025858, −0.10120781433427948519655163143,
0.10120781433427948519655163143, 1.21454759450132784891243025858, 1.48987807716648405614549165875, 2.80181462649680150735487422175, 3.12373145515759498350568064050, 4.05343344979342324821249028099, 4.70422700344555341511780918719, 4.98500003939528906685647104033, 5.84865193661912456967108777962, 6.27630639084589703059708576624, 6.76468939624162810992345830483, 7.58307450135194912509226140097, 7.66655130267807684623141202925, 8.508014197101937328426509615105, 8.586118616181049151805917464997, 9.536254148590375068907934523476, 9.565308963540641386559321365447, 10.26816884732463667571960251537, 10.69130732031347884104465603253, 11.30579771906466616373457624666
