| L(s) = 1 | + 0.523·2-s − 3-s − 1.72·4-s − 5-s − 0.523·6-s − 3.44·7-s − 1.95·8-s + 9-s − 0.523·10-s + 3.03·11-s + 1.72·12-s − 5.69·13-s − 1.80·14-s + 15-s + 2.42·16-s + 3.41·17-s + 0.523·18-s − 4.93·19-s + 1.72·20-s + 3.44·21-s + 1.59·22-s + 1.95·24-s + 25-s − 2.98·26-s − 27-s + 5.94·28-s + 5.89·29-s + ⋯ |
| L(s) = 1 | + 0.370·2-s − 0.577·3-s − 0.862·4-s − 0.447·5-s − 0.213·6-s − 1.30·7-s − 0.690·8-s + 0.333·9-s − 0.165·10-s + 0.916·11-s + 0.498·12-s − 1.57·13-s − 0.482·14-s + 0.258·15-s + 0.607·16-s + 0.828·17-s + 0.123·18-s − 1.13·19-s + 0.385·20-s + 0.751·21-s + 0.339·22-s + 0.398·24-s + 0.200·25-s − 0.584·26-s − 0.192·27-s + 1.12·28-s + 1.09·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.523T + 2T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 3.03T + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 + 4.93T + 19T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 - 4.05T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 - 6.25T + 47T^{2} \) |
| 53 | \( 1 + 2.95T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 - 8.01T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 7.14T + 89T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
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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
−7.31152388006184647231205248083, −6.64097947592584627309073829896, −6.18134041943506852255511490713, −5.26693777540156716863611109847, −4.68942160750395644364700164501, −3.95813360901480893952606165977, −3.35983433147672737800481345288, −2.44239969899365411091540071655, −0.866021346442349947534790298194, 0,
0.866021346442349947534790298194, 2.44239969899365411091540071655, 3.35983433147672737800481345288, 3.95813360901480893952606165977, 4.68942160750395644364700164501, 5.26693777540156716863611109847, 6.18134041943506852255511490713, 6.64097947592584627309073829896, 7.31152388006184647231205248083
