| L(s) = 1 | + 0.712·2-s − 3-s − 1.49·4-s − 0.712·6-s − 2.77·7-s − 2.48·8-s + 9-s + 4.26·11-s + 1.49·12-s + 0.779·13-s − 1.98·14-s + 1.21·16-s − 1.90·17-s + 0.712·18-s + 6.72·19-s + 2.77·21-s + 3.04·22-s + 2.17·23-s + 2.48·24-s + 0.555·26-s − 27-s + 4.14·28-s + 29-s − 8.82·31-s + 5.83·32-s − 4.26·33-s − 1.35·34-s + ⋯ |
| L(s) = 1 | + 0.503·2-s − 0.577·3-s − 0.746·4-s − 0.290·6-s − 1.05·7-s − 0.879·8-s + 0.333·9-s + 1.28·11-s + 0.430·12-s + 0.216·13-s − 0.529·14-s + 0.302·16-s − 0.461·17-s + 0.167·18-s + 1.54·19-s + 0.606·21-s + 0.648·22-s + 0.452·23-s + 0.507·24-s + 0.108·26-s − 0.192·27-s + 0.783·28-s + 0.185·29-s − 1.58·31-s + 1.03·32-s − 0.742·33-s − 0.232·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.712T + 2T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 - 0.779T + 13T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 + 7.71T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 4.46T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 8.49T + 67T^{2} \) |
| 71 | \( 1 - 0.663T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 9.54T + 79T^{2} \) |
| 83 | \( 1 + 0.0123T + 83T^{2} \) |
| 89 | \( 1 + 5.46T + 89T^{2} \) |
| 97 | \( 1 + 0.952T + 97T^{2} \) |
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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
−9.015825693527396108120868020832, −7.85950359589051773324718604151, −6.74257224241768851914314453122, −6.35621466555676673674208619209, −5.40533470545678818905974101564, −4.73401512552929285281975518636, −3.64577480205658030495224175197, −3.24936797193772198072486306143, −1.38585558305090380304521523032, 0,
1.38585558305090380304521523032, 3.24936797193772198072486306143, 3.64577480205658030495224175197, 4.73401512552929285281975518636, 5.40533470545678818905974101564, 6.35621466555676673674208619209, 6.74257224241768851914314453122, 7.85950359589051773324718604151, 9.015825693527396108120868020832
