| L(s) = 1 | + 1.30·2-s + 0.334·3-s − 0.302·4-s + 0.436·6-s + 3.90·7-s − 2.99·8-s − 2.88·9-s − 0.135·11-s − 0.101·12-s − 6.65·13-s + 5.08·14-s − 3.30·16-s − 3.76·18-s − 5.16·19-s + 1.30·21-s − 0.176·22-s + 4.32·23-s − 1.00·24-s − 8.67·26-s − 1.97·27-s − 1.18·28-s − 2.14·29-s + 8.61·31-s + 1.69·32-s − 0.0453·33-s + 0.873·36-s + 0.234·37-s + ⋯ |
| L(s) = 1 | + 0.921·2-s + 0.193·3-s − 0.151·4-s + 0.178·6-s + 1.47·7-s − 1.06·8-s − 0.962·9-s − 0.0408·11-s − 0.0292·12-s − 1.84·13-s + 1.35·14-s − 0.825·16-s − 0.886·18-s − 1.18·19-s + 0.285·21-s − 0.0376·22-s + 0.901·23-s − 0.204·24-s − 1.70·26-s − 0.379·27-s − 0.223·28-s − 0.398·29-s + 1.54·31-s + 0.299·32-s − 0.00789·33-s + 0.145·36-s + 0.0385·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.571378693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.571378693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 0.334T + 3T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 + 0.135T + 11T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 + 2.14T + 29T^{2} \) |
| 31 | \( 1 - 8.61T + 31T^{2} \) |
| 37 | \( 1 - 0.234T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 1.89T + 43T^{2} \) |
| 47 | \( 1 - 9.86T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 8.10T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 1.26T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 + 4.30T + 89T^{2} \) |
| 97 | \( 1 + 0.473T + 97T^{2} \) |
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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
−7.966728621024898697513123689607, −7.25875699376531620879299883270, −6.32363734023026436246715301019, −5.55693312709260858433327089189, −4.91031319722251061385475076273, −4.59421317548806032145256235744, −3.73981605282311589289469828327, −2.53247656946161313461513806648, −2.35417698448044852656091346311, −0.67362848496811495240874602182,
0.67362848496811495240874602182, 2.35417698448044852656091346311, 2.53247656946161313461513806648, 3.73981605282311589289469828327, 4.59421317548806032145256235744, 4.91031319722251061385475076273, 5.55693312709260858433327089189, 6.32363734023026436246715301019, 7.25875699376531620879299883270, 7.966728621024898697513123689607
