| L(s) = 1 | − 0.728·2-s − 2.86·3-s − 1.46·4-s + 5-s + 2.08·6-s + 1.10·7-s + 2.52·8-s + 5.23·9-s − 0.728·10-s + 1.34·11-s + 4.21·12-s + 2.88·13-s − 0.805·14-s − 2.86·15-s + 1.09·16-s + 0.885·17-s − 3.81·18-s + 5.92·19-s − 1.46·20-s − 3.17·21-s − 0.976·22-s + 7.33·23-s − 7.25·24-s + 25-s − 2.10·26-s − 6.41·27-s − 1.62·28-s + ⋯ |
| L(s) = 1 | − 0.514·2-s − 1.65·3-s − 0.734·4-s + 0.447·5-s + 0.853·6-s + 0.418·7-s + 0.893·8-s + 1.74·9-s − 0.230·10-s + 0.404·11-s + 1.21·12-s + 0.800·13-s − 0.215·14-s − 0.740·15-s + 0.274·16-s + 0.214·17-s − 0.898·18-s + 1.35·19-s − 0.328·20-s − 0.692·21-s − 0.208·22-s + 1.53·23-s − 1.47·24-s + 0.200·25-s − 0.412·26-s − 1.23·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7939613236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7939613236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.728T + 2T^{2} \) |
| 3 | \( 1 + 2.86T + 3T^{2} \) |
| 7 | \( 1 - 1.10T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 0.885T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 0.0300T + 37T^{2} \) |
| 41 | \( 1 + 0.301T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 9.47T + 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 8.25T + 67T^{2} \) |
| 71 | \( 1 + 4.88T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−9.244915010342901491970127712413, −8.562157228076359356015655310505, −7.40930611217606522330180426080, −6.86761485209197660241616731756, −5.62651346928815858304519908874, −5.42561692074147203477815820906, −4.53089121729521757556259521584, −3.53138566214482761419398471080, −1.52739355258719959673963091195, −0.790820682101554316364636724321,
0.790820682101554316364636724321, 1.52739355258719959673963091195, 3.53138566214482761419398471080, 4.53089121729521757556259521584, 5.42561692074147203477815820906, 5.62651346928815858304519908874, 6.86761485209197660241616731756, 7.40930611217606522330180426080, 8.562157228076359356015655310505, 9.244915010342901491970127712413
