| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 8-s − 10-s + 5·11-s + 10·13-s − 15-s − 16-s − 4·17-s − 7·19-s − 5·22-s − 23-s − 24-s − 10·26-s + 27-s + 30-s − 2·31-s − 5·33-s + 4·34-s − 37-s + 7·38-s − 10·39-s + 40-s − 10·41-s + 24·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 0.353·8-s − 0.316·10-s + 1.50·11-s + 2.77·13-s − 0.258·15-s − 1/4·16-s − 0.970·17-s − 1.60·19-s − 1.06·22-s − 0.208·23-s − 0.204·24-s − 1.96·26-s + 0.192·27-s + 0.182·30-s − 0.359·31-s − 0.870·33-s + 0.685·34-s − 0.164·37-s + 1.13·38-s − 1.60·39-s + 0.158·40-s − 1.56·41-s + 3.65·43-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
| \(L(1)\) |
\(\approx\) |
\(1.634010229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634010229\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 11 T + 74 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.435413652965648313187719772280, −9.295540633667486553488677637269, −8.791737987112281731247924016495, −8.771450324102540866398319110555, −8.197063417870288484938099167247, −7.962443689353808899390997020851, −7.18129045237025270709502669029, −6.66259239372151572736551898856, −6.35665203711728041128358800612, −6.28784629431410555132294326749, −5.81845873209178267374812243227, −5.17787118292393023773268653176, −4.69336199533490211124736815670, −3.92594625838563655550085650694, −3.88286022290860931007901230824, −3.46912508939379291925484888133, −2.15224662097179625024985562554, −2.08686137011015217623153341833, −1.14721591490270083247109626258, −0.71917019474342350466976607913,
0.71917019474342350466976607913, 1.14721591490270083247109626258, 2.08686137011015217623153341833, 2.15224662097179625024985562554, 3.46912508939379291925484888133, 3.88286022290860931007901230824, 3.92594625838563655550085650694, 4.69336199533490211124736815670, 5.17787118292393023773268653176, 5.81845873209178267374812243227, 6.28784629431410555132294326749, 6.35665203711728041128358800612, 6.66259239372151572736551898856, 7.18129045237025270709502669029, 7.962443689353808899390997020851, 8.197063417870288484938099167247, 8.771450324102540866398319110555, 8.791737987112281731247924016495, 9.295540633667486553488677637269, 9.435413652965648313187719772280
