| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 2.44·11-s + 4.44·13-s + 15-s − 5.44·17-s − 4.44·19-s − 21-s + 23-s + 25-s − 27-s + 10.3·29-s − 0.101·31-s + 2.44·33-s − 35-s + 3.89·37-s − 4.44·39-s + 5.44·41-s − 2·43-s − 45-s − 8.44·47-s − 6·49-s + 5.44·51-s + 0.550·53-s + 2.44·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 0.738·11-s + 1.23·13-s + 0.258·15-s − 1.32·17-s − 1.02·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.92·29-s − 0.0181·31-s + 0.426·33-s − 0.169·35-s + 0.640·37-s − 0.712·39-s + 0.851·41-s − 0.304·43-s − 0.149·45-s − 1.23·47-s − 0.857·49-s + 0.763·51-s + 0.0756·53-s + 0.330·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 0.101T + 31T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 8.44T + 47T^{2} \) |
| 53 | \( 1 - 0.550T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 0.651T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 - 4.34T + 71T^{2} \) |
| 73 | \( 1 + 5.34T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 3.10T + 97T^{2} \) |
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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
−7.998900351165692221970364504029, −6.88614684630696585502663221729, −6.43319724873137140849601587825, −5.71496757545350526494324552262, −4.60763332796836225500383153260, −4.45170956927122151966525697776, −3.29316303795388708183035422721, −2.32756786166233649443248476252, −1.21209491126946288057364593563, 0,
1.21209491126946288057364593563, 2.32756786166233649443248476252, 3.29316303795388708183035422721, 4.45170956927122151966525697776, 4.60763332796836225500383153260, 5.71496757545350526494324552262, 6.43319724873137140849601587825, 6.88614684630696585502663221729, 7.998900351165692221970364504029
