| L(s) = 1 | + 0.830·2-s + 3-s − 0.309·4-s + 1.68·5-s + 0.830·6-s − 1.91·7-s − 1.08·8-s + 9-s + 1.39·10-s − 0.284·11-s − 0.309·12-s − 1.59·14-s + 1.68·15-s − 0.594·16-s − 1.30·17-s + 0.830·18-s − 0.284·19-s − 0.521·20-s − 1.91·21-s − 0.236·22-s − 1.08·24-s + 1.83·25-s + 27-s + 0.594·28-s + 1.39·30-s + 0.594·32-s − 0.284·33-s + ⋯ |
| L(s) = 1 | + 0.830·2-s + 3-s − 0.309·4-s + 1.68·5-s + 0.830·6-s − 1.91·7-s − 1.08·8-s + 9-s + 1.39·10-s − 0.284·11-s − 0.309·12-s − 1.59·14-s + 1.68·15-s − 0.594·16-s − 1.30·17-s + 0.830·18-s − 0.284·19-s − 0.521·20-s − 1.91·21-s − 0.236·22-s − 1.08·24-s + 1.83·25-s + 27-s + 0.594·28-s + 1.39·30-s + 0.594·32-s − 0.284·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 591 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 591 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.638183834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638183834\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 - 0.830T + T^{2} \) |
| 5 | \( 1 - 1.68T + T^{2} \) |
| 7 | \( 1 + 1.91T + T^{2} \) |
| 11 | \( 1 + 0.284T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.30T + T^{2} \) |
| 19 | \( 1 + 0.284T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.30T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.830T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.68T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.91T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.830T + T^{2} \) |
| 97 | \( 1 + 1.30T + T^{2} \) |
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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
−10.49150906248885486083687311459, −9.849664842716384708429602615371, −9.187027736851609049896725021662, −8.734003908887180001366307832634, −6.91055500580376906521053595545, −6.32919895151901999340199458956, −5.42369569191795396236866110225, −4.15553796664477471072882470712, −3.08302653378897140921481583928, −2.31264171519099436316359024194,
2.31264171519099436316359024194, 3.08302653378897140921481583928, 4.15553796664477471072882470712, 5.42369569191795396236866110225, 6.32919895151901999340199458956, 6.91055500580376906521053595545, 8.734003908887180001366307832634, 9.187027736851609049896725021662, 9.849664842716384708429602615371, 10.49150906248885486083687311459
