| L(s) = 1 | + 1.92·2-s + 1.71·4-s + 0.200·5-s + 3.71·7-s − 0.545·8-s + 0.387·10-s − 1.10·11-s + 0.253·13-s + 7.15·14-s − 4.48·16-s + 3.55·17-s + 0.345·20-s − 2.12·22-s + 7.35·23-s − 4.95·25-s + 0.487·26-s + 6.37·28-s − 1.86·29-s + 4.12·31-s − 7.55·32-s + 6.86·34-s + 0.745·35-s + 7.85·37-s − 0.109·40-s + 2.79·41-s + 2.42·43-s − 1.89·44-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.858·4-s + 0.0898·5-s + 1.40·7-s − 0.193·8-s + 0.122·10-s − 0.332·11-s + 0.0701·13-s + 1.91·14-s − 1.12·16-s + 0.863·17-s + 0.0771·20-s − 0.453·22-s + 1.53·23-s − 0.991·25-s + 0.0956·26-s + 1.20·28-s − 0.347·29-s + 0.740·31-s − 1.33·32-s + 1.17·34-s + 0.126·35-s + 1.29·37-s − 0.0173·40-s + 0.437·41-s + 0.369·43-s − 0.285·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.418620401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.418620401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 5 | \( 1 - 0.200T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 0.253T + 13T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 23 | \( 1 - 7.35T + 23T^{2} \) |
| 29 | \( 1 + 1.86T + 29T^{2} \) |
| 31 | \( 1 - 4.12T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 5.70T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 + 0.700T + 71T^{2} \) |
| 73 | \( 1 - 0.889T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−7.65702106323740957686260746266, −6.88553471828624076699398839517, −5.89455459686466571057456866435, −5.60995275791605413890954172036, −4.71620331537709591279423319783, −4.50419646055964875719216590242, −3.52632381016622582928776515140, −2.79874272276272439687064349757, −1.98197913707749288271097853744, −0.930755535648229456026034830937,
0.930755535648229456026034830937, 1.98197913707749288271097853744, 2.79874272276272439687064349757, 3.52632381016622582928776515140, 4.50419646055964875719216590242, 4.71620331537709591279423319783, 5.60995275791605413890954172036, 5.89455459686466571057456866435, 6.88553471828624076699398839517, 7.65702106323740957686260746266
