| L(s) = 1 | − 6·3-s + 2·4-s + 6·5-s − 8·7-s + 27·9-s + 66·11-s − 12·12-s − 2·13-s − 36·15-s − 60·16-s − 34·17-s − 26·19-s + 12·20-s + 48·21-s − 198·23-s + 65·25-s − 108·27-s − 16·28-s + 444·29-s + 532·31-s − 396·33-s − 48·35-s + 54·36-s + 88·37-s + 12·39-s + 570·41-s − 182·43-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/4·4-s + 0.536·5-s − 0.431·7-s + 9-s + 1.80·11-s − 0.288·12-s − 0.0426·13-s − 0.619·15-s − 0.937·16-s − 0.485·17-s − 0.313·19-s + 0.134·20-s + 0.498·21-s − 1.79·23-s + 0.519·25-s − 0.769·27-s − 0.107·28-s + 2.84·29-s + 3.08·31-s − 2.08·33-s − 0.231·35-s + 1/4·36-s + 0.391·37-s + 0.0492·39-s + 2.17·41-s − 0.645·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.328492758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328492758\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 6 T - 29 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 414 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 p T + 3463 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 3243 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 26 T + 9279 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 198 T + 33847 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 444 T + 93454 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 532 T + 129186 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 88 T + 29514 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 570 T + 195739 T^{2} - 570 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 182 T + 162687 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 420 T + 249154 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 300 T + 43486 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 444 T + 294154 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 400 T + 55914 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 968 T + 742470 T^{2} + 968 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1848 T + 1513150 T^{2} + 1848 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 868 T + 963798 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 p T + 1388986 T^{2} - 12 p^{4} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 288 T + 287602 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1024 T + 2038818 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−15.61319356534030693898734410656, −14.62362983344960776055261691856, −14.05702462900541792188703586250, −13.68222288785837112621247610811, −12.93585579326445167328565546576, −12.06647203562279580501175738999, −11.90615907726882415022258997630, −11.48140757846097856424454487758, −10.35686539836264492321707011189, −10.27469207882236894398521211578, −9.408194808190971945735719944340, −8.800583935613334014062643720068, −7.894159518313359336398061507610, −6.64262873624525573446634557869, −6.42113878209371864504508109495, −6.11885099356846956181751603264, −4.61604693073622875217041840227, −4.27935599618588121820624358010, −2.55702699647835084381266876097, −1.01793237515708370202812005337,
1.01793237515708370202812005337, 2.55702699647835084381266876097, 4.27935599618588121820624358010, 4.61604693073622875217041840227, 6.11885099356846956181751603264, 6.42113878209371864504508109495, 6.64262873624525573446634557869, 7.894159518313359336398061507610, 8.800583935613334014062643720068, 9.408194808190971945735719944340, 10.27469207882236894398521211578, 10.35686539836264492321707011189, 11.48140757846097856424454487758, 11.90615907726882415022258997630, 12.06647203562279580501175738999, 12.93585579326445167328565546576, 13.68222288785837112621247610811, 14.05702462900541792188703586250, 14.62362983344960776055261691856, 15.61319356534030693898734410656
