| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 4·8-s + 3·9-s − 4·10-s − 4·11-s − 6·12-s − 4·15-s + 5·16-s + 8·17-s − 6·18-s − 8·19-s + 6·20-s + 8·22-s + 12·23-s + 8·24-s + 3·25-s − 4·27-s + 8·29-s + 8·30-s − 12·31-s − 6·32-s + 8·33-s − 16·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s − 1.20·11-s − 1.73·12-s − 1.03·15-s + 5/4·16-s + 1.94·17-s − 1.41·18-s − 1.83·19-s + 1.34·20-s + 1.70·22-s + 2.50·23-s + 1.63·24-s + 3/5·25-s − 0.769·27-s + 1.48·29-s + 1.46·30-s − 2.15·31-s − 1.06·32-s + 1.39·33-s − 2.74·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001489990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001489990\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.733285241697315604767741879971, −9.280149518025642141406665033699, −8.959184756505256443497317599205, −8.759756208564014220552681014969, −7.977809534325356318101921351504, −7.70435008066193203384493680290, −7.35019609921314411283933631029, −7.02347788447525799477078133697, −6.29792174281127491510778634348, −6.18125565339828675029212663643, −5.77461662748256950288469817078, −5.21833765073136983848455326884, −4.95211788084688748619227352142, −4.42192514583588912725900418514, −3.43687336418807623598169474559, −3.06859717288621484091304238475, −2.29712506192564374505929072677, −1.97581342947159375154038880819, −0.942803363561806052611621428449, −0.72516733990553923785407835706,
0.72516733990553923785407835706, 0.942803363561806052611621428449, 1.97581342947159375154038880819, 2.29712506192564374505929072677, 3.06859717288621484091304238475, 3.43687336418807623598169474559, 4.42192514583588912725900418514, 4.95211788084688748619227352142, 5.21833765073136983848455326884, 5.77461662748256950288469817078, 6.18125565339828675029212663643, 6.29792174281127491510778634348, 7.02347788447525799477078133697, 7.35019609921314411283933631029, 7.70435008066193203384493680290, 7.977809534325356318101921351504, 8.759756208564014220552681014969, 8.959184756505256443497317599205, 9.280149518025642141406665033699, 9.733285241697315604767741879971
