L(s) = 1 | − 3-s − 3.92·5-s + 7-s − 2·9-s − 2·11-s + 4.92·13-s + 3.92·15-s + 17-s + 1.92·19-s − 21-s − 6.10·23-s + 10.3·25-s + 5·27-s − 5.92·29-s + 3.73·31-s + 2·33-s − 3.92·35-s + 8.65·37-s − 4.92·39-s + 1.92·41-s − 0.266·43-s + 7.84·45-s + 2.10·47-s − 6·49-s − 51-s + 9.37·53-s + 7.84·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.75·5-s + 0.377·7-s − 0.666·9-s − 0.603·11-s + 1.36·13-s + 1.01·15-s + 0.242·17-s + 0.440·19-s − 0.218·21-s − 1.27·23-s + 2.07·25-s + 0.962·27-s − 1.09·29-s + 0.670·31-s + 0.348·33-s − 0.662·35-s + 1.42·37-s − 0.787·39-s + 0.300·41-s − 0.0405·43-s + 1.16·45-s + 0.307·47-s − 0.857·49-s − 0.140·51-s + 1.28·53-s + 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.92T + 13T^{2} \) |
| 19 | \( 1 - 1.92T + 19T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 0.266T + 43T^{2} \) |
| 47 | \( 1 - 2.10T + 47T^{2} \) |
| 53 | \( 1 - 9.37T + 53T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 4.65T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 + 3.45T + 97T^{2} \) |
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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
−8.029543041451572794945177254801, −7.61534479522816023643667154472, −6.56623538986048203928823539720, −5.82995478223402936263624602935, −5.08936022576443553844333274072, −4.14900301099168479685349612341, −3.62503438546142810655571113065, −2.64576878800819781500371666205, −1.05882803055996315281064102656, 0,
1.05882803055996315281064102656, 2.64576878800819781500371666205, 3.62503438546142810655571113065, 4.14900301099168479685349612341, 5.08936022576443553844333274072, 5.82995478223402936263624602935, 6.56623538986048203928823539720, 7.61534479522816023643667154472, 8.029543041451572794945177254801