| L(s) = 1 | + 2-s − 2.67·3-s + 4-s − 5-s − 2.67·6-s + 0.990·7-s + 8-s + 4.15·9-s − 10-s − 3.19·11-s − 2.67·12-s + 1.76·13-s + 0.990·14-s + 2.67·15-s + 16-s − 3.31·17-s + 4.15·18-s + 0.673·19-s − 20-s − 2.65·21-s − 3.19·22-s + 2.71·23-s − 2.67·24-s + 25-s + 1.76·26-s − 3.10·27-s + 0.990·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.54·3-s + 0.5·4-s − 0.447·5-s − 1.09·6-s + 0.374·7-s + 0.353·8-s + 1.38·9-s − 0.316·10-s − 0.963·11-s − 0.772·12-s + 0.489·13-s + 0.264·14-s + 0.690·15-s + 0.250·16-s − 0.803·17-s + 0.980·18-s + 0.154·19-s − 0.223·20-s − 0.578·21-s − 0.681·22-s + 0.566·23-s − 0.546·24-s + 0.200·25-s + 0.346·26-s − 0.596·27-s + 0.187·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 2.67T + 3T^{2} \) |
| 7 | \( 1 - 0.990T + 7T^{2} \) |
| 11 | \( 1 + 3.19T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 - 0.673T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 1.88T + 31T^{2} \) |
| 37 | \( 1 + 2.33T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 + 7.24T + 53T^{2} \) |
| 59 | \( 1 - 8.12T + 59T^{2} \) |
| 61 | \( 1 - 5.88T + 61T^{2} \) |
| 67 | \( 1 + 9.12T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−7.934846630221907727038795914456, −6.97166027349093224532565262151, −6.58211754559781205641119227712, −5.64232042565263122256187148959, −5.12033113620074387152269478980, −4.56856292515990975015817801082, −3.66399324079712885309547273836, −2.54681619154558301097024768079, −1.27080034839403260274910062598, 0,
1.27080034839403260274910062598, 2.54681619154558301097024768079, 3.66399324079712885309547273836, 4.56856292515990975015817801082, 5.12033113620074387152269478980, 5.64232042565263122256187148959, 6.58211754559781205641119227712, 6.97166027349093224532565262151, 7.934846630221907727038795914456
