L(s) = 1 | − 2.71·3-s + 0.521·5-s + 4.38·9-s − 0.0523·11-s + 5.95·13-s − 1.41·15-s − 8.06·17-s − 4.41·19-s + 5.28·23-s − 4.72·25-s − 3.76·27-s + 4.73·29-s + 8.03·31-s + 0.142·33-s − 5.28·37-s − 16.1·39-s − 41-s + 3.65·43-s + 2.28·45-s − 4.87·47-s + 21.9·51-s − 1.13·53-s − 0.0273·55-s + 11.9·57-s − 5.21·59-s − 10.2·61-s + 3.10·65-s + ⋯ |
L(s) = 1 | − 1.56·3-s + 0.233·5-s + 1.46·9-s − 0.0157·11-s + 1.65·13-s − 0.365·15-s − 1.95·17-s − 1.01·19-s + 1.10·23-s − 0.945·25-s − 0.725·27-s + 0.879·29-s + 1.44·31-s + 0.0247·33-s − 0.868·37-s − 2.59·39-s − 0.156·41-s + 0.557·43-s + 0.340·45-s − 0.710·47-s + 3.06·51-s − 0.155·53-s − 0.00368·55-s + 1.58·57-s − 0.679·59-s − 1.31·61-s + 0.385·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 - 0.521T + 5T^{2} \) |
| 11 | \( 1 + 0.0523T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 + 8.06T + 17T^{2} \) |
| 19 | \( 1 + 4.41T + 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 8.03T + 31T^{2} \) |
| 37 | \( 1 + 5.28T + 37T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 + 4.87T + 47T^{2} \) |
| 53 | \( 1 + 1.13T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 - 9.25T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + 6.66T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
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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
−7.13158452980921428855694536734, −6.47543800516061978475692381657, −6.25768069870924709350867258598, −5.55128263040304285177057178556, −4.53985431263904882641377155097, −4.37212266278320910293990026851, −3.14985900308355988221027396721, −1.98609619005362648591970505689, −1.06856883179536101224375088155, 0,
1.06856883179536101224375088155, 1.98609619005362648591970505689, 3.14985900308355988221027396721, 4.37212266278320910293990026851, 4.53985431263904882641377155097, 5.55128263040304285177057178556, 6.25768069870924709350867258598, 6.47543800516061978475692381657, 7.13158452980921428855694536734