| L(s) = 1 | + 2.41·2-s − 0.537·3-s + 3.85·4-s − 1.29·6-s + 3.18·7-s + 4.49·8-s − 2.71·9-s + 4.15·11-s − 2.07·12-s − 2.07·13-s + 7.71·14-s + 3.15·16-s − 5.79·17-s − 6.56·18-s − 19-s − 1.71·21-s + 10.0·22-s + 2.60·23-s − 2.41·24-s − 5.01·26-s + 3.06·27-s + 12.2·28-s + 6·29-s + 2.59·31-s − 1.34·32-s − 2.23·33-s − 14.0·34-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 0.310·3-s + 1.92·4-s − 0.530·6-s + 1.20·7-s + 1.58·8-s − 0.903·9-s + 1.25·11-s − 0.597·12-s − 0.574·13-s + 2.06·14-s + 0.788·16-s − 1.40·17-s − 1.54·18-s − 0.229·19-s − 0.373·21-s + 2.14·22-s + 0.543·23-s − 0.492·24-s − 0.982·26-s + 0.590·27-s + 2.32·28-s + 1.11·29-s + 0.466·31-s − 0.237·32-s − 0.388·33-s − 2.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.456026870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.456026870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 + 0.537T + 3T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 + 4.30T + 37T^{2} \) |
| 41 | \( 1 + 0.599T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 2.72T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 2.07T + 97T^{2} \) |
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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
−11.42914549877807920452444824913, −10.72963131554115289684461111801, −9.125612398491009934797537683860, −8.161361446199647684270679948434, −6.77589657797894483872649228642, −6.23747713927914624437019652328, −4.91908694619561831623350060779, −4.61746627410580203861480523527, −3.23585636369928182955420610606, −1.93948977006853465589612621317,
1.93948977006853465589612621317, 3.23585636369928182955420610606, 4.61746627410580203861480523527, 4.91908694619561831623350060779, 6.23747713927914624437019652328, 6.77589657797894483872649228642, 8.161361446199647684270679948434, 9.125612398491009934797537683860, 10.72963131554115289684461111801, 11.42914549877807920452444824913
