| L(s) = 1 | + 5-s − 4·7-s + 3·11-s − 4·13-s + 5·19-s − 6·23-s + 25-s − 9·29-s + 5·31-s − 4·35-s + 2·37-s − 9·41-s − 10·43-s − 6·47-s + 9·49-s − 12·53-s + 3·55-s + 9·59-s − 10·61-s − 4·65-s + 2·67-s + 3·71-s − 4·73-s − 12·77-s − 4·79-s + 6·83-s − 9·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 0.904·11-s − 1.10·13-s + 1.14·19-s − 1.25·23-s + 1/5·25-s − 1.67·29-s + 0.898·31-s − 0.676·35-s + 0.328·37-s − 1.40·41-s − 1.52·43-s − 0.875·47-s + 9/7·49-s − 1.64·53-s + 0.404·55-s + 1.17·59-s − 1.28·61-s − 0.496·65-s + 0.244·67-s + 0.356·71-s − 0.468·73-s − 1.36·77-s − 0.450·79-s + 0.658·83-s − 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.365735209724249650801766208602, −8.221440095252496236140880696261, −7.22071661878271177619872980253, −6.56690061153787637882102352572, −5.87248703622753539329170166212, −4.90523607973036056720187714059, −3.71162973193978748507760260210, −2.99055288852233283223860215491, −1.73691204456417348658756664708, 0,
1.73691204456417348658756664708, 2.99055288852233283223860215491, 3.71162973193978748507760260210, 4.90523607973036056720187714059, 5.87248703622753539329170166212, 6.56690061153787637882102352572, 7.22071661878271177619872980253, 8.221440095252496236140880696261, 9.365735209724249650801766208602
