L(s) = 1 | − 1.21·3-s − 5-s + 7-s − 1.52·9-s − 5.76·11-s − 13-s + 1.21·15-s + 3.42·17-s + 3.23·19-s − 1.21·21-s + 9.23·23-s + 25-s + 5.49·27-s + 6.64·29-s + 1.84·31-s + 7.00·33-s − 35-s + 2.07·37-s + 1.21·39-s − 12.0·41-s − 4.52·43-s + 1.52·45-s + 1.33·47-s + 49-s − 4.16·51-s − 5.05·53-s + 5.76·55-s + ⋯ |
L(s) = 1 | − 0.702·3-s − 0.447·5-s + 0.377·7-s − 0.507·9-s − 1.73·11-s − 0.277·13-s + 0.313·15-s + 0.830·17-s + 0.743·19-s − 0.265·21-s + 1.92·23-s + 0.200·25-s + 1.05·27-s + 1.23·29-s + 0.330·31-s + 1.21·33-s − 0.169·35-s + 0.340·37-s + 0.194·39-s − 1.88·41-s − 0.690·43-s + 0.226·45-s + 0.194·47-s + 0.142·49-s − 0.582·51-s − 0.694·53-s + 0.776·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 1.21T + 3T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 9.23T + 23T^{2} \) |
| 29 | \( 1 - 6.64T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 4.52T + 43T^{2} \) |
| 47 | \( 1 - 1.33T + 47T^{2} \) |
| 53 | \( 1 + 5.05T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 + 9.00T + 61T^{2} \) |
| 67 | \( 1 + 7.28T + 67T^{2} \) |
| 71 | \( 1 - 6.33T + 71T^{2} \) |
| 73 | \( 1 + 0.186T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 9.15T + 89T^{2} \) |
| 97 | \( 1 + 8.37T + 97T^{2} \) |
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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
−8.126976873893232307467757723342, −7.48078378936289933274007398180, −6.73446182281860852496215126690, −5.74134524941966911857121253886, −5.01317339926538442264785952417, −4.80414553659645810738698178801, −3.17474189809578312520743219721, −2.80047521280895651640945899625, −1.18576167267202214895480562939, 0,
1.18576167267202214895480562939, 2.80047521280895651640945899625, 3.17474189809578312520743219721, 4.80414553659645810738698178801, 5.01317339926538442264785952417, 5.74134524941966911857121253886, 6.73446182281860852496215126690, 7.48078378936289933274007398180, 8.126976873893232307467757723342