| L(s) = 1 | − 5-s + 7-s − 5.12·11-s + 2·13-s − 5.12·17-s + 7.12·19-s + 5.12·23-s + 25-s + 3.12·29-s − 10.2·31-s − 35-s − 1.12·37-s + 8.24·41-s + 7.12·43-s + 7.12·47-s + 49-s − 7.12·53-s + 5.12·55-s + 4·59-s + 2.87·61-s − 2·65-s + 7.12·67-s + 6·71-s + 12.2·73-s − 5.12·77-s + 8·79-s + 5.12·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.54·11-s + 0.554·13-s − 1.24·17-s + 1.63·19-s + 1.06·23-s + 0.200·25-s + 0.579·29-s − 1.84·31-s − 0.169·35-s − 0.184·37-s + 1.28·41-s + 1.08·43-s + 1.03·47-s + 0.142·49-s − 0.978·53-s + 0.690·55-s + 0.520·59-s + 0.368·61-s − 0.248·65-s + 0.870·67-s + 0.712·71-s + 1.43·73-s − 0.583·77-s + 0.900·79-s + 0.555·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.517038722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517038722\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.935056490710658837804115580360, −8.013071364253420400222228263790, −7.51072172722384432481978817080, −6.77036492838624777266103360244, −5.56206150202373014074066517657, −5.10245176336731106296550126953, −4.12019398219651811164067422073, −3.13094762100683885740917396152, −2.23990488551356045429248242990, −0.77525889949161483816020627592,
0.77525889949161483816020627592, 2.23990488551356045429248242990, 3.13094762100683885740917396152, 4.12019398219651811164067422073, 5.10245176336731106296550126953, 5.56206150202373014074066517657, 6.77036492838624777266103360244, 7.51072172722384432481978817080, 8.013071364253420400222228263790, 8.935056490710658837804115580360
