| L(s) = 1 | − 1.73·2-s − 3.39·3-s + 1.01·4-s − 2.86·5-s + 5.89·6-s − 0.942·7-s + 1.71·8-s + 8.53·9-s + 4.96·10-s − 3.33·11-s − 3.43·12-s − 1.74·13-s + 1.63·14-s + 9.71·15-s − 4.99·16-s + 3.59·17-s − 14.8·18-s + 1.93·19-s − 2.89·20-s + 3.20·21-s + 5.78·22-s − 2.66·23-s − 5.82·24-s + 3.18·25-s + 3.03·26-s − 18.8·27-s − 0.953·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 1.96·3-s + 0.505·4-s − 1.27·5-s + 2.40·6-s − 0.356·7-s + 0.606·8-s + 2.84·9-s + 1.56·10-s − 1.00·11-s − 0.992·12-s − 0.484·13-s + 0.437·14-s + 2.50·15-s − 1.24·16-s + 0.871·17-s − 3.49·18-s + 0.445·19-s − 0.647·20-s + 0.698·21-s + 1.23·22-s − 0.554·23-s − 1.18·24-s + 0.636·25-s + 0.594·26-s − 3.61·27-s − 0.180·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2016254818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2016254818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 3.39T + 3T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 + 0.942T + 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 - 0.721T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 - 2.72T + 53T^{2} \) |
| 59 | \( 1 - 7.83T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 - 8.28T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 - 0.714T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
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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
−7.71564955521326025658327874453, −7.24095884554476792210325937267, −6.52408829536772909243250742652, −5.57690240178839528697621063331, −5.14942609115994114408212832881, −4.28581863699472716220608575935, −3.72869572665651264020294697520, −2.24972910387399941351578612396, −0.964412248783771735165777400906, −0.39382894998196682103665782673,
0.39382894998196682103665782673, 0.964412248783771735165777400906, 2.24972910387399941351578612396, 3.72869572665651264020294697520, 4.28581863699472716220608575935, 5.14942609115994114408212832881, 5.57690240178839528697621063331, 6.52408829536772909243250742652, 7.24095884554476792210325937267, 7.71564955521326025658327874453
