| L(s) = 1 | − 3-s − 5-s − 1.52·7-s + 9-s + 3.59·11-s + 5.59·13-s + 15-s + 4.07·17-s + 1.59·19-s + 1.52·21-s − 23-s + 25-s − 27-s − 4.07·29-s − 2.47·31-s − 3.59·33-s + 1.52·35-s + 9.66·37-s − 5.59·39-s + 5.01·41-s − 7.05·43-s − 45-s + 4.54·47-s − 4.66·49-s − 4.07·51-s + 5.01·53-s − 3.59·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.576·7-s + 0.333·9-s + 1.08·11-s + 1.55·13-s + 0.258·15-s + 0.987·17-s + 0.366·19-s + 0.333·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.756·29-s − 0.444·31-s − 0.626·33-s + 0.258·35-s + 1.58·37-s − 0.896·39-s + 0.783·41-s − 1.07·43-s − 0.149·45-s + 0.663·47-s − 0.667·49-s − 0.570·51-s + 0.689·53-s − 0.485·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628224080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628224080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 + 7.05T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 - 5.01T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 + 8.79T + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.107895723541288185326195039099, −7.37955776434057439861707872423, −6.62398934185607041365153503565, −6.01254943833673489252234234595, −5.49001830752502589657451499027, −4.29056757281591785483366264030, −3.77661811621993423556503932306, −3.07016059682869520306203526823, −1.57806176310532293795507537439, −0.76080912806436791773259687589,
0.76080912806436791773259687589, 1.57806176310532293795507537439, 3.07016059682869520306203526823, 3.77661811621993423556503932306, 4.29056757281591785483366264030, 5.49001830752502589657451499027, 6.01254943833673489252234234595, 6.62398934185607041365153503565, 7.37955776434057439861707872423, 8.107895723541288185326195039099
