| L(s) = 1 | + 5-s + 7-s − 2·13-s − 2·19-s + 25-s − 6·29-s − 4·31-s + 35-s − 4·37-s + 9·41-s + 43-s + 3·47-s − 6·49-s + 6·53-s + 61-s − 2·65-s − 13·67-s + 12·71-s + 16·73-s + 10·79-s + 12·83-s + 3·89-s − 2·91-s − 2·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.169·35-s − 0.657·37-s + 1.40·41-s + 0.152·43-s + 0.437·47-s − 6/7·49-s + 0.824·53-s + 0.128·61-s − 0.248·65-s − 1.58·67-s + 1.42·71-s + 1.87·73-s + 1.12·79-s + 1.31·83-s + 0.317·89-s − 0.209·91-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.77520486044845, −15.19456706423662, −14.63614107787269, −14.36427463311601, −13.57025532189280, −13.22260707133271, −12.42418753840507, −12.21544336528580, −11.32161229860587, −10.85587034116314, −10.44481959755627, −9.524696175575973, −9.341781420855724, −8.618219850963325, −7.874772563932360, −7.462201722196920, −6.716360800910322, −6.146017895892479, −5.362501897359892, −5.022741926423458, −4.109693558703558, −3.570450408315346, −2.520756543858095, −2.059413484816205, −1.149055069014068, 0,
1.149055069014068, 2.059413484816205, 2.520756543858095, 3.570450408315346, 4.109693558703558, 5.022741926423458, 5.362501897359892, 6.146017895892479, 6.716360800910322, 7.462201722196920, 7.874772563932360, 8.618219850963325, 9.341781420855724, 9.524696175575973, 10.44481959755627, 10.85587034116314, 11.32161229860587, 12.21544336528580, 12.42418753840507, 13.22260707133271, 13.57025532189280, 14.36427463311601, 14.63614107787269, 15.19456706423662, 15.77520486044845
