| 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 − 568·11-s + 1.03e4·12-s − 1.48e4·13-s + 1.09e4·14-s + 1.35e4·15-s + 2.04e4·16-s + 2.26e3·17-s − 3.49e4·18-s + 1.07e4·19-s + 4.80e4·20-s − 3.70e4·21-s + 9.08e3·22-s + 6.24e4·23-s − 1.10e5·24-s + 4.68e4·25-s + 2.37e5·26-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 − 0.128·11-s + 1.73·12-s − 1.87·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.111·17-s − 1.41·18-s + 0.360·19-s + 1.34·20-s − 0.872·21-s + 0.181·22-s + 1.07·23-s − 1.63·24-s + 3/5·25-s + 2.64·26-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
| \(L(4)\) |
\(\approx\) |
\(3.437784883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.437784883\) |
| \(L(\frac{9}{2})\) |
|
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 | $D_{4}$ | \( 1 + 568 T - 3156586 T^{2} + 568 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1140 p T + 169852238 T^{2} + 1140 p^{8} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2260 T - 32830330 T^{2} - 2260 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10776 T + 1647927686 T^{2} - 10776 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 62496 T + 7268996494 T^{2} - 62496 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 172380 T + 37274641582 T^{2} - 172380 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 229168 T + 62572239294 T^{2} - 229168 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 356812 T + 208765157502 T^{2} - 356812 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 884788 T + 544655666774 T^{2} - 884788 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 237352 T - 128480847210 T^{2} + 237352 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 280208 T + 873220608158 T^{2} + 280208 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1834948 T + 3068091841406 T^{2} + 1834948 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2060056 T + 5641091856566 T^{2} + 2060056 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1038908 T + 4312606119342 T^{2} - 1038908 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 732120 T + 6461028445670 T^{2} + 732120 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2041024 T + 12290823748910 T^{2} - 2041024 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4345828 T + 14936735651574 T^{2} - 4345828 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6300896 T + 37094053063518 T^{2} - 6300896 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15843544 T + 116487573883142 T^{2} - 15843544 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 913748 T + 87195306762038 T^{2} - 913748 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10709140 T + 179609300058726 T^{2} - 10709140 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.96178774788190251164565384807, −10.66592291142676413407943594529, −9.979893124480786557263644135941, −9.745531345686352767504045988400, −9.309388028052536179010579386952, −9.246632568227713622039083511033, −8.350559617894875845959445320639, −7.960560324354484611117582415891, −7.48377367346434246463882702340, −6.99930605680816530721358712476, −6.38408984461968118863418058539, −6.05441860900186835311874879498, −4.86970039199387452635410859629, −4.68280408547993136090399785283, −3.24483067109270944999755813575, −3.02475632085726671225880380557, −2.29727391615320443163451158835, −2.07130580871687831360422845636, −0.854556612010641922897301434845, −0.70308761044010547221413621283,
0.70308761044010547221413621283, 0.854556612010641922897301434845, 2.07130580871687831360422845636, 2.29727391615320443163451158835, 3.02475632085726671225880380557, 3.24483067109270944999755813575, 4.68280408547993136090399785283, 4.86970039199387452635410859629, 6.05441860900186835311874879498, 6.38408984461968118863418058539, 6.99930605680816530721358712476, 7.48377367346434246463882702340, 7.960560324354484611117582415891, 8.350559617894875845959445320639, 9.246632568227713622039083511033, 9.309388028052536179010579386952, 9.745531345686352767504045988400, 9.979893124480786557263644135941, 10.66592291142676413407943594529, 10.96178774788190251164565384807
