| L(s) = 1 | + 2.56·3-s + 5-s − 7-s + 3.56·9-s + 2.56·11-s − 0.561·13-s + 2.56·15-s + 4.56·17-s − 2.56·21-s + 25-s + 1.43·27-s − 5.68·29-s + 5.12·31-s + 6.56·33-s − 35-s + 7.12·37-s − 1.43·39-s + 2·41-s + 4·43-s + 3.56·45-s − 11.6·47-s + 49-s + 11.6·51-s + 4.87·53-s + 2.56·55-s − 10.2·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.447·5-s − 0.377·7-s + 1.18·9-s + 0.772·11-s − 0.155·13-s + 0.661·15-s + 1.10·17-s − 0.558·21-s + 0.200·25-s + 0.276·27-s − 1.05·29-s + 0.920·31-s + 1.14·33-s − 0.169·35-s + 1.17·37-s − 0.230·39-s + 0.312·41-s + 0.609·43-s + 0.530·45-s − 1.70·47-s + 0.142·49-s + 1.63·51-s + 0.669·53-s + 0.345·55-s − 1.33·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.946848808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.946848808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−9.510922785690889493197895011216, −9.223019265299746810967712706600, −8.189520472897882244799224614392, −7.58799332587869740191750876370, −6.58593964586480863824770921193, −5.66034790784514080468657297455, −4.33698396760612261870592622093, −3.41612325477367864016092879734, −2.62460941525072255993409601957, −1.43916284198260404977767765669,
1.43916284198260404977767765669, 2.62460941525072255993409601957, 3.41612325477367864016092879734, 4.33698396760612261870592622093, 5.66034790784514080468657297455, 6.58593964586480863824770921193, 7.58799332587869740191750876370, 8.189520472897882244799224614392, 9.223019265299746810967712706600, 9.510922785690889493197895011216
