| L(s) = 1 | + 2-s − 3-s + 5-s − 6-s + 7-s − 8-s + 10-s + 12·11-s + 13-s + 14-s − 15-s − 16-s − 6·17-s + 7·19-s − 21-s + 12·22-s − 12·23-s + 24-s + 26-s + 27-s + 18·29-s − 30-s − 8·31-s − 12·33-s − 6·34-s + 35-s + 11·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 3.61·11-s + 0.277·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 1.60·19-s − 0.218·21-s + 2.55·22-s − 2.50·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 3.34·29-s − 0.182·30-s − 1.43·31-s − 2.08·33-s − 1.02·34-s + 0.169·35-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.778425431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.778425431\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 37 | $C_2$ | \( 1 - 11 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.953871990082091940938851236081, −9.541927689555402402623443515630, −9.228378603034700844243454519974, −9.020732830746475835283969620751, −8.322048092597783534463145228437, −8.274585891953708611016338931558, −7.15395611354107512078693749426, −7.09264296644029755282107444231, −6.54609075013246584393561701979, −6.15886860605771446854277883946, −5.80399760247020061614618320794, −5.63168780548413473117476710773, −4.60502796469245766931162250838, −4.28906806490066852181619308797, −4.15348208551456601175354539965, −3.64903469629494670609858691602, −2.82786302841431761784719495373, −2.16321645654747343765217437596, −1.35268290978761980872789518326, −0.939947349923226685331139817729,
0.939947349923226685331139817729, 1.35268290978761980872789518326, 2.16321645654747343765217437596, 2.82786302841431761784719495373, 3.64903469629494670609858691602, 4.15348208551456601175354539965, 4.28906806490066852181619308797, 4.60502796469245766931162250838, 5.63168780548413473117476710773, 5.80399760247020061614618320794, 6.15886860605771446854277883946, 6.54609075013246584393561701979, 7.09264296644029755282107444231, 7.15395611354107512078693749426, 8.274585891953708611016338931558, 8.322048092597783534463145228437, 9.020732830746475835283969620751, 9.228378603034700844243454519974, 9.541927689555402402623443515630, 9.953871990082091940938851236081
