| L(s) = 1 | − 2.53·2-s − 3.30·3-s + 4.44·4-s − 5-s + 8.38·6-s + 3.67·7-s − 6.21·8-s + 7.91·9-s + 2.53·10-s + 4.92·11-s − 14.6·12-s + 4.53·13-s − 9.32·14-s + 3.30·15-s + 6.87·16-s − 2.84·17-s − 20.0·18-s + 1.00·19-s − 4.44·20-s − 12.1·21-s − 12.5·22-s + 5.54·23-s + 20.5·24-s + 25-s − 11.5·26-s − 16.2·27-s + 16.3·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.90·3-s + 2.22·4-s − 0.447·5-s + 3.42·6-s + 1.38·7-s − 2.19·8-s + 2.63·9-s + 0.802·10-s + 1.48·11-s − 4.24·12-s + 1.25·13-s − 2.49·14-s + 0.852·15-s + 1.71·16-s − 0.690·17-s − 4.73·18-s + 0.229·19-s − 0.994·20-s − 2.64·21-s − 2.66·22-s + 1.15·23-s + 4.18·24-s + 0.200·25-s − 2.25·26-s − 3.12·27-s + 3.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4617946424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4617946424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 - 5.54T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 + 7.90T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 3.70T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 0.165T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 + 4.01T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 7.18T + 97T^{2} \) |
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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
−10.62757950093979928048820009958, −9.522484183787753657519027229701, −8.687712007887429559176709915534, −7.82641798315581090500095176308, −6.79129711290227615978475629760, −6.42702275385608811303885272726, −5.16205149582402837225565583397, −4.08659842031388009078930060242, −1.55899442967525151151957241998, −0.925099179888259101015846931606,
0.925099179888259101015846931606, 1.55899442967525151151957241998, 4.08659842031388009078930060242, 5.16205149582402837225565583397, 6.42702275385608811303885272726, 6.79129711290227615978475629760, 7.82641798315581090500095176308, 8.687712007887429559176709915534, 9.522484183787753657519027229701, 10.62757950093979928048820009958
