| L(s) = 1 | − 1.24·2-s − 3-s − 0.457·4-s + 2.60·5-s + 1.24·6-s + 3.18·7-s + 3.05·8-s + 9-s − 3.23·10-s + 0.457·12-s + 4.38·13-s − 3.96·14-s − 2.60·15-s − 2.87·16-s − 2.32·17-s − 1.24·18-s − 19-s − 1.19·20-s − 3.18·21-s + 5.10·23-s − 3.05·24-s + 1.77·25-s − 5.44·26-s − 27-s − 1.45·28-s − 4.68·29-s + 3.23·30-s + ⋯ |
| L(s) = 1 | − 0.878·2-s − 0.577·3-s − 0.228·4-s + 1.16·5-s + 0.507·6-s + 1.20·7-s + 1.07·8-s + 0.333·9-s − 1.02·10-s + 0.131·12-s + 1.21·13-s − 1.05·14-s − 0.672·15-s − 0.719·16-s − 0.564·17-s − 0.292·18-s − 0.229·19-s − 0.266·20-s − 0.696·21-s + 1.06·23-s − 0.622·24-s + 0.355·25-s − 1.06·26-s − 0.192·27-s − 0.275·28-s − 0.870·29-s + 0.590·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.495145251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495145251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 + 2.32T + 17T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 + 5.72T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 0.884T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 0.502T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 - 5.78T + 67T^{2} \) |
| 71 | \( 1 - 9.87T + 71T^{2} \) |
| 73 | \( 1 + 4.11T + 73T^{2} \) |
| 79 | \( 1 + 5.49T + 79T^{2} \) |
| 83 | \( 1 - 8.18T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 8.91T + 97T^{2} \) |
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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
−7.998460240618112853983682811288, −7.45227012786514249391977166606, −6.54851446502266425138170618463, −5.82671844454170145253209740204, −5.20973212447626381979475774964, −4.55240486403264590211212627406, −3.73258312282949051328069021678, −2.20669705463134794795077275348, −1.58548964961317996553872403064, −0.809040933770763821414515387145,
0.809040933770763821414515387145, 1.58548964961317996553872403064, 2.20669705463134794795077275348, 3.73258312282949051328069021678, 4.55240486403264590211212627406, 5.20973212447626381979475774964, 5.82671844454170145253209740204, 6.54851446502266425138170618463, 7.45227012786514249391977166606, 7.998460240618112853983682811288
