| L(s) = 1 | − 0.345·3-s + 2.40·5-s − 1.76·7-s − 2.88·9-s − 2.38·11-s + 4.86·13-s − 0.832·15-s + 4.36·17-s − 4.69·19-s + 0.610·21-s − 1.71·23-s + 0.799·25-s + 2.03·27-s + 2.59·29-s − 0.727·31-s + 0.824·33-s − 4.25·35-s − 7.06·37-s − 1.68·39-s − 0.505·41-s + 5.98·43-s − 6.93·45-s − 8.49·47-s − 3.88·49-s − 1.50·51-s + 4.93·53-s − 5.74·55-s + ⋯ |
| L(s) = 1 | − 0.199·3-s + 1.07·5-s − 0.667·7-s − 0.960·9-s − 0.718·11-s + 1.34·13-s − 0.214·15-s + 1.05·17-s − 1.07·19-s + 0.133·21-s − 0.356·23-s + 0.159·25-s + 0.391·27-s + 0.482·29-s − 0.130·31-s + 0.143·33-s − 0.718·35-s − 1.16·37-s − 0.269·39-s − 0.0789·41-s + 0.912·43-s − 1.03·45-s − 1.23·47-s − 0.554·49-s − 0.211·51-s + 0.678·53-s − 0.774·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 0.345T + 3T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 - 4.86T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 - 2.59T + 29T^{2} \) |
| 31 | \( 1 + 0.727T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + 0.505T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 + 8.49T + 47T^{2} \) |
| 53 | \( 1 - 4.93T + 53T^{2} \) |
| 59 | \( 1 + 7.80T + 59T^{2} \) |
| 61 | \( 1 - 6.09T + 61T^{2} \) |
| 67 | \( 1 + 0.910T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 7.63T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.88841624648194322354784755661, −6.72000746125402245559554326851, −6.21303404673421379405357049404, −5.68173592751908298717170875008, −5.14092729791772968078439515758, −3.91668490639326121875542716921, −3.14551955983788669850201388221, −2.37640658032411415920476645294, −1.36302550940283962618178374990, 0,
1.36302550940283962618178374990, 2.37640658032411415920476645294, 3.14551955983788669850201388221, 3.91668490639326121875542716921, 5.14092729791772968078439515758, 5.68173592751908298717170875008, 6.21303404673421379405357049404, 6.72000746125402245559554326851, 7.88841624648194322354784755661
