| L(s) = 1 | − 1.64·2-s − 2.98·3-s + 0.719·4-s + 5-s + 4.91·6-s − 5.09·7-s + 2.11·8-s + 5.88·9-s − 1.64·10-s + 11-s − 2.14·12-s − 6.30·13-s + 8.39·14-s − 2.98·15-s − 4.92·16-s + 6.73·17-s − 9.70·18-s − 19-s + 0.719·20-s + 15.1·21-s − 1.64·22-s + 4.03·23-s − 6.29·24-s + 25-s + 10.4·26-s − 8.58·27-s − 3.66·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 1.72·3-s + 0.359·4-s + 0.447·5-s + 2.00·6-s − 1.92·7-s + 0.746·8-s + 1.96·9-s − 0.521·10-s + 0.301·11-s − 0.619·12-s − 1.74·13-s + 2.24·14-s − 0.769·15-s − 1.23·16-s + 1.63·17-s − 2.28·18-s − 0.229·19-s + 0.160·20-s + 3.31·21-s − 0.351·22-s + 0.841·23-s − 1.28·24-s + 0.200·25-s + 2.04·26-s − 1.65·27-s − 0.692·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 7 | \( 1 + 5.09T + 7T^{2} \) |
| 13 | \( 1 + 6.30T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 - 4.01T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 3.05T + 67T^{2} \) |
| 71 | \( 1 - 4.13T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 1.16T + 83T^{2} \) |
| 89 | \( 1 + 0.0957T + 89T^{2} \) |
| 97 | \( 1 - 3.15T + 97T^{2} \) |
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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
−9.789587692315674443437947997482, −9.119303607299377365698736842593, −7.62290838650624812373405262110, −6.91839086472044642343709639050, −6.29223402169452722602874050372, −5.38434085464934130977777870701, −4.47827978029679645494208611259, −2.88779136754228324942297818325, −1.05219134737681470428961781473, 0,
1.05219134737681470428961781473, 2.88779136754228324942297818325, 4.47827978029679645494208611259, 5.38434085464934130977777870701, 6.29223402169452722602874050372, 6.91839086472044642343709639050, 7.62290838650624812373405262110, 9.119303607299377365698736842593, 9.789587692315674443437947997482
