| L(s) = 1 | + 2.30·2-s − 3-s + 3.32·4-s − 2.30·6-s − 0.860·7-s + 3.06·8-s + 9-s + 2.86·11-s − 3.32·12-s − 4.01·13-s − 1.98·14-s + 0.412·16-s − 6.25·17-s + 2.30·18-s − 7.86·19-s + 0.860·21-s + 6.62·22-s − 1.67·23-s − 3.06·24-s − 9.27·26-s − 27-s − 2.86·28-s − 1.38·29-s + 7.32·31-s − 5.17·32-s − 2.86·33-s − 14.4·34-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.577·3-s + 1.66·4-s − 0.942·6-s − 0.325·7-s + 1.08·8-s + 0.333·9-s + 0.865·11-s − 0.960·12-s − 1.11·13-s − 0.530·14-s + 0.103·16-s − 1.51·17-s + 0.543·18-s − 1.80·19-s + 0.187·21-s + 1.41·22-s − 0.349·23-s − 0.624·24-s − 1.81·26-s − 0.192·27-s − 0.540·28-s − 0.256·29-s + 1.31·31-s − 0.914·32-s − 0.499·33-s − 2.47·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 7 | \( 1 + 0.860T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 + 6.25T + 17T^{2} \) |
| 19 | \( 1 + 7.86T + 19T^{2} \) |
| 23 | \( 1 + 1.67T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 7.32T + 31T^{2} \) |
| 37 | \( 1 + 7.22T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 43 | \( 1 - 6.88T + 43T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 8.17T + 59T^{2} \) |
| 61 | \( 1 + 7.18T + 61T^{2} \) |
| 67 | \( 1 - 5.56T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 6.31T + 83T^{2} \) |
| 89 | \( 1 + 8.21T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.985795635872553451488851914807, −6.86633915290861095413089830836, −6.51525781088553956316079998505, −6.00365946984107042041217879946, −4.90140295447385000547381480824, −4.46336193903261822714065417908, −3.83510523907487483007775972351, −2.66653341685097017271951044601, −1.92900633913268504650010876655, 0,
1.92900633913268504650010876655, 2.66653341685097017271951044601, 3.83510523907487483007775972351, 4.46336193903261822714065417908, 4.90140295447385000547381480824, 6.00365946984107042041217879946, 6.51525781088553956316079998505, 6.86633915290861095413089830836, 7.985795635872553451488851914807
