| L(s) = 1 | − 2·7-s + 3·11-s − 2·13-s + 3·17-s − 19-s − 6·23-s − 5·25-s + 6·29-s + 4·31-s + 4·37-s − 9·41-s − 43-s − 6·47-s − 3·49-s + 12·53-s − 3·59-s − 8·61-s + 5·67-s − 12·71-s + 11·73-s − 6·77-s + 4·79-s − 12·83-s − 6·89-s + 4·91-s + 5·97-s − 14·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.904·11-s − 0.554·13-s + 0.727·17-s − 0.229·19-s − 1.25·23-s − 25-s + 1.11·29-s + 0.718·31-s + 0.657·37-s − 1.40·41-s − 0.152·43-s − 0.875·47-s − 3/7·49-s + 1.64·53-s − 0.390·59-s − 1.02·61-s + 0.610·67-s − 1.42·71-s + 1.28·73-s − 0.683·77-s + 0.450·79-s − 1.31·83-s − 0.635·89-s + 0.419·91-s + 0.507·97-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.930769578371338402756573985837, −7.03958314562926506761932301326, −6.38621009962525321192846394616, −5.86722110566278992511976681176, −4.86353492334106993174795804756, −4.05064804668395326172520142038, −3.35055085317237919419053143292, −2.42466756648709547485696737515, −1.34471597583379685537750572425, 0,
1.34471597583379685537750572425, 2.42466756648709547485696737515, 3.35055085317237919419053143292, 4.05064804668395326172520142038, 4.86353492334106993174795804756, 5.86722110566278992511976681176, 6.38621009962525321192846394616, 7.03958314562926506761932301326, 7.930769578371338402756573985837
