| L(s) = 1 | + 0.661·2-s − 3.01·3-s − 1.56·4-s − 1.99·6-s − 0.592·7-s − 2.35·8-s + 6.06·9-s + 11-s + 4.70·12-s − 3.68·13-s − 0.391·14-s + 1.56·16-s − 0.251·17-s + 4.01·18-s + 19-s + 1.78·21-s + 0.661·22-s − 2.58·23-s + 7.09·24-s − 2.43·26-s − 9.23·27-s + 0.925·28-s + 2.22·29-s − 7.50·31-s + 5.74·32-s − 3.01·33-s − 0.165·34-s + ⋯ |
| L(s) = 1 | + 0.467·2-s − 1.73·3-s − 0.781·4-s − 0.812·6-s − 0.223·7-s − 0.832·8-s + 2.02·9-s + 0.301·11-s + 1.35·12-s − 1.02·13-s − 0.104·14-s + 0.391·16-s − 0.0608·17-s + 0.945·18-s + 0.229·19-s + 0.389·21-s + 0.140·22-s − 0.538·23-s + 1.44·24-s − 0.477·26-s − 1.77·27-s + 0.174·28-s + 0.413·29-s − 1.34·31-s + 1.01·32-s − 0.524·33-s − 0.0284·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3952773419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3952773419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.661T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 7 | \( 1 + 0.592T + 7T^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 + 0.251T + 17T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 + 9.64T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 0.283T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 3.89T + 53T^{2} \) |
| 59 | \( 1 + 2.98T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 7.91T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 3.74T + 83T^{2} \) |
| 89 | \( 1 + 3.78T + 89T^{2} \) |
| 97 | \( 1 + 9.18T + 97T^{2} \) |
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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
−8.083339575599360537732558914106, −7.19120694846189650630676143914, −6.61117501704924426302182694956, −5.73982756734886390906459427879, −5.41203531210925676702031842432, −4.62559314489582609857112867657, −4.11105731157139108735271577584, −3.08298758211028263746753311538, −1.61803163078866726729260489873, −0.35313140513760119402846866746,
0.35313140513760119402846866746, 1.61803163078866726729260489873, 3.08298758211028263746753311538, 4.11105731157139108735271577584, 4.62559314489582609857112867657, 5.41203531210925676702031842432, 5.73982756734886390906459427879, 6.61117501704924426302182694956, 7.19120694846189650630676143914, 8.083339575599360537732558914106
