| L(s) = 1 | − 2.78·3-s + 3.12·5-s + 4.93·7-s + 4.78·9-s + 2.03·11-s + 4.58·13-s − 8.73·15-s − 17-s + 4.73·19-s − 13.7·21-s − 1.83·23-s + 4.79·25-s − 4.97·27-s − 4.03·29-s + 4.92·31-s − 5.67·33-s + 15.4·35-s − 10.0·37-s − 12.7·39-s − 2.17·41-s + 6.41·43-s + 14.9·45-s + 12.5·47-s + 17.3·49-s + 2.78·51-s + 10.9·53-s + 6.36·55-s + ⋯ |
| L(s) = 1 | − 1.61·3-s + 1.39·5-s + 1.86·7-s + 1.59·9-s + 0.613·11-s + 1.27·13-s − 2.25·15-s − 0.242·17-s + 1.08·19-s − 3.00·21-s − 0.383·23-s + 0.958·25-s − 0.956·27-s − 0.749·29-s + 0.884·31-s − 0.987·33-s + 2.60·35-s − 1.64·37-s − 2.04·39-s − 0.339·41-s + 0.977·43-s + 2.23·45-s + 1.83·47-s + 2.47·49-s + 0.390·51-s + 1.50·53-s + 0.858·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.549319025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549319025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 - 4.93T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 61 | \( 1 + 8.91T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 9.14T + 89T^{2} \) |
| 97 | \( 1 + 2.28T + 97T^{2} \) |
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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
−7.63129070988749101635346308523, −6.99346048448610225521646865228, −6.16288823787913593352450794811, −5.66832414809582632537130868947, −5.32109819010775469773912414975, −4.56921104647926173067572281478, −3.81861879419577125761942092400, −2.27042694698743009581548856164, −1.43305971634513778068683966135, −1.03241846506138147463362467464,
1.03241846506138147463362467464, 1.43305971634513778068683966135, 2.27042694698743009581548856164, 3.81861879419577125761942092400, 4.56921104647926173067572281478, 5.32109819010775469773912414975, 5.66832414809582632537130868947, 6.16288823787913593352450794811, 6.99346048448610225521646865228, 7.63129070988749101635346308523
