L(s) = 1 | + 2.35·2-s + 3.55·4-s − 0.424·5-s + 0.792·7-s + 3.66·8-s − 10-s − 0.527·13-s + 1.86·14-s + 1.52·16-s − 4.51·17-s − 3.34·19-s − 1.50·20-s + 6.02·23-s − 4.81·25-s − 1.24·26-s + 2.81·28-s − 9.82·29-s − 0.942·31-s − 3.72·32-s − 10.6·34-s − 0.336·35-s − 8.40·37-s − 7.88·38-s − 1.55·40-s + 3.47·41-s + 3.49·43-s + 14.1·46-s + ⋯ |
L(s) = 1 | + 1.66·2-s + 1.77·4-s − 0.189·5-s + 0.299·7-s + 1.29·8-s − 0.316·10-s − 0.146·13-s + 0.499·14-s + 0.381·16-s − 1.09·17-s − 0.767·19-s − 0.337·20-s + 1.25·23-s − 0.963·25-s − 0.243·26-s + 0.532·28-s − 1.82·29-s − 0.169·31-s − 0.659·32-s − 1.82·34-s − 0.0568·35-s − 1.38·37-s − 1.27·38-s − 0.245·40-s + 0.542·41-s + 0.533·43-s + 2.09·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 + 0.424T + 5T^{2} \) |
| 7 | \( 1 - 0.792T + 7T^{2} \) |
| 13 | \( 1 + 0.527T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + 9.82T + 29T^{2} \) |
| 31 | \( 1 + 0.942T + 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 + 3.69T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 3.00T + 83T^{2} \) |
| 89 | \( 1 - 8.43T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−7.18622461035497142736770610831, −6.39018609133357001852682541847, −5.92009714453589251453008777074, −4.98003836530275916065999232521, −4.72496317820611553293285813930, −3.80778191140329349147638789110, −3.36222689908294538894412320213, −2.30919654660437840926800670632, −1.76048282797948837946297829012, 0,
1.76048282797948837946297829012, 2.30919654660437840926800670632, 3.36222689908294538894412320213, 3.80778191140329349147638789110, 4.72496317820611553293285813930, 4.98003836530275916065999232521, 5.92009714453589251453008777074, 6.39018609133357001852682541847, 7.18622461035497142736770610831