L(s) = 1 | + 0.445·2-s − 3-s − 1.80·4-s − 3.55·5-s − 0.445·6-s + 7-s − 1.69·8-s + 9-s − 1.58·10-s − 1.60·11-s + 1.80·12-s + 0.445·14-s + 3.55·15-s + 2.85·16-s − 0.0554·17-s + 0.445·18-s + 4.41·19-s + 6.41·20-s − 21-s − 0.715·22-s − 7.96·23-s + 1.69·24-s + 7.66·25-s − 27-s − 1.80·28-s + 7.43·29-s + 1.58·30-s + ⋯ |
L(s) = 1 | + 0.314·2-s − 0.577·3-s − 0.900·4-s − 1.59·5-s − 0.181·6-s + 0.377·7-s − 0.598·8-s + 0.333·9-s − 0.500·10-s − 0.484·11-s + 0.520·12-s + 0.118·14-s + 0.919·15-s + 0.712·16-s − 0.0134·17-s + 0.104·18-s + 1.01·19-s + 1.43·20-s − 0.218·21-s − 0.152·22-s − 1.66·23-s + 0.345·24-s + 1.53·25-s − 0.192·27-s − 0.340·28-s + 1.38·29-s + 0.289·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 17 | \( 1 + 0.0554T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 + 7.96T + 23T^{2} \) |
| 29 | \( 1 - 7.43T + 29T^{2} \) |
| 31 | \( 1 - 9.55T + 31T^{2} \) |
| 37 | \( 1 - 8.02T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 8.88T + 43T^{2} \) |
| 47 | \( 1 + 7.76T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 + 3.04T + 59T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 - 9.63T + 67T^{2} \) |
| 71 | \( 1 + 2.05T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 7.13T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.125581579438375459640348527318, −7.70064884777606566389120561458, −6.66377727191937413336580700920, −5.81124977399041352395251483197, −4.85237782876445172865143778801, −4.49568574321382640826226008541, −3.71672625877769576805524070685, −2.84878118764335873001951521140, −1.02374925907169832148184554924, 0,
1.02374925907169832148184554924, 2.84878118764335873001951521140, 3.71672625877769576805524070685, 4.49568574321382640826226008541, 4.85237782876445172865143778801, 5.81124977399041352395251483197, 6.66377727191937413336580700920, 7.70064884777606566389120561458, 8.125581579438375459640348527318