| L(s) = 1 | − 2.37·3-s + 5-s + 2.37·7-s + 2.62·9-s + 11-s + 2·13-s − 2.37·15-s − 4.37·17-s − 6.37·19-s − 5.62·21-s + 8.74·23-s + 25-s + 0.883·27-s − 4.37·29-s + 2.37·31-s − 2.37·33-s + 2.37·35-s + 3.62·37-s − 4.74·39-s + 11.4·41-s + 4·43-s + 2.62·45-s + 8.74·47-s − 1.37·49-s + 10.3·51-s + 13.1·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.447·5-s + 0.896·7-s + 0.875·9-s + 0.301·11-s + 0.554·13-s − 0.612·15-s − 1.06·17-s − 1.46·19-s − 1.22·21-s + 1.82·23-s + 0.200·25-s + 0.169·27-s − 0.811·29-s + 0.426·31-s − 0.412·33-s + 0.400·35-s + 0.596·37-s − 0.759·39-s + 1.79·41-s + 0.609·43-s + 0.391·45-s + 1.27·47-s − 0.196·49-s + 1.45·51-s + 1.80·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.157809246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157809246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 - 8.74T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−10.61752351708743171375610804488, −9.229156629510218857885998521868, −8.626162617092044418344673357684, −7.37396894341638042029095593713, −6.46252711868132526935783001627, −5.84850725987968490518089300873, −4.89165278209256050949436076297, −4.18819722131704446756572370402, −2.33678020520911131616529515445, −0.962152251017449576302150756695,
0.962152251017449576302150756695, 2.33678020520911131616529515445, 4.18819722131704446756572370402, 4.89165278209256050949436076297, 5.84850725987968490518089300873, 6.46252711868132526935783001627, 7.37396894341638042029095593713, 8.626162617092044418344673357684, 9.229156629510218857885998521868, 10.61752351708743171375610804488
