| L(s) = 1 | + 0.925·2-s − 1.14·4-s − 3.88·5-s + 1.90·7-s − 2.90·8-s − 3.59·10-s − 0.783·11-s − 6.58·13-s + 1.76·14-s − 0.403·16-s − 7.22·17-s − 19-s + 4.44·20-s − 0.724·22-s − 2.25·23-s + 10.0·25-s − 6.09·26-s − 2.18·28-s − 8.64·29-s + 1.84·31-s + 5.44·32-s − 6.68·34-s − 7.40·35-s − 6.85·37-s − 0.925·38-s + 11.2·40-s − 12.0·41-s + ⋯ |
| L(s) = 1 | + 0.654·2-s − 0.571·4-s − 1.73·5-s + 0.721·7-s − 1.02·8-s − 1.13·10-s − 0.236·11-s − 1.82·13-s + 0.471·14-s − 0.100·16-s − 1.75·17-s − 0.229·19-s + 0.992·20-s − 0.154·22-s − 0.471·23-s + 2.01·25-s − 1.19·26-s − 0.412·28-s − 1.60·29-s + 0.331·31-s + 0.962·32-s − 1.14·34-s − 1.25·35-s − 1.12·37-s − 0.150·38-s + 1.78·40-s − 1.88·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.001484929617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001484929617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.925T + 2T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 11 | \( 1 + 0.783T + 11T^{2} \) |
| 13 | \( 1 + 6.58T + 13T^{2} \) |
| 17 | \( 1 + 7.22T + 17T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 8.64T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 53 | \( 1 - 2.03T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 - 5.69T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 - 3.81T + 89T^{2} \) |
| 97 | \( 1 + 7.19T + 97T^{2} \) |
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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
−7.81135487527765978826660559477, −7.17714642752378071946328334094, −6.60559660728495556567588185495, −5.26787066003203336424744984787, −4.95325375776725219152085169662, −4.28913169918998991927546400964, −3.80619985248327968317504533391, −2.90833120697688224570886126625, −1.94893888013080307584177495036, −0.01622720458745393286854797297,
0.01622720458745393286854797297, 1.94893888013080307584177495036, 2.90833120697688224570886126625, 3.80619985248327968317504533391, 4.28913169918998991927546400964, 4.95325375776725219152085169662, 5.26787066003203336424744984787, 6.60559660728495556567588185495, 7.17714642752378071946328334094, 7.81135487527765978826660559477
