| L(s) = 1 | − 2.49·2-s − 0.867·3-s + 4.24·4-s + 3.27·5-s + 2.16·6-s + 1.98·7-s − 5.60·8-s − 2.24·9-s − 8.18·10-s + 2.44·11-s − 3.68·12-s − 4.07·13-s − 4.95·14-s − 2.84·15-s + 5.51·16-s − 3.48·17-s + 5.61·18-s + 1.20·19-s + 13.8·20-s − 1.71·21-s − 6.11·22-s − 5.75·23-s + 4.86·24-s + 5.72·25-s + 10.1·26-s + 4.55·27-s + 8.40·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.500·3-s + 2.12·4-s + 1.46·5-s + 0.885·6-s + 0.748·7-s − 1.98·8-s − 0.749·9-s − 2.58·10-s + 0.737·11-s − 1.06·12-s − 1.13·13-s − 1.32·14-s − 0.733·15-s + 1.37·16-s − 0.844·17-s + 1.32·18-s + 0.277·19-s + 3.10·20-s − 0.375·21-s − 1.30·22-s − 1.19·23-s + 0.992·24-s + 1.14·25-s + 1.99·26-s + 0.876·27-s + 1.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + 0.867T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 0.892T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 - 9.25T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 5.36T + 89T^{2} \) |
| 97 | \( 1 - 0.827T + 97T^{2} \) |
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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
−8.216468135669841468981312400741, −7.52570492517361785748857954601, −6.65115506488089424166189051160, −6.11280307896131736029426575900, −5.42866172578260973369265907939, −4.45727788944385718167323388687, −2.72764018556800629302384821952, −2.08309209435185629236457287714, −1.32949943305441162398605276227, 0,
1.32949943305441162398605276227, 2.08309209435185629236457287714, 2.72764018556800629302384821952, 4.45727788944385718167323388687, 5.42866172578260973369265907939, 6.11280307896131736029426575900, 6.65115506488089424166189051160, 7.52570492517361785748857954601, 8.216468135669841468981312400741
