| L(s) = 1 | + 3-s − 5-s − 3.63·7-s + 9-s − 5.52·11-s + 0.707·13-s − 15-s − 1.04·17-s + 0.542·19-s − 3.63·21-s + 5.02·23-s + 25-s + 27-s + 9.47·29-s − 6.10·31-s − 5.52·33-s + 3.63·35-s − 1.83·37-s + 0.707·39-s − 6.32·41-s + 6.20·43-s − 45-s − 8.96·47-s + 6.23·49-s − 1.04·51-s + 6.97·53-s + 5.52·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.37·7-s + 0.333·9-s − 1.66·11-s + 0.196·13-s − 0.258·15-s − 0.253·17-s + 0.124·19-s − 0.793·21-s + 1.04·23-s + 0.200·25-s + 0.192·27-s + 1.75·29-s − 1.09·31-s − 0.961·33-s + 0.614·35-s − 0.302·37-s + 0.113·39-s − 0.987·41-s + 0.945·43-s − 0.149·45-s − 1.30·47-s + 0.891·49-s − 0.146·51-s + 0.958·53-s + 0.744·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.306060623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306060623\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 - 0.707T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 0.542T + 19T^{2} \) |
| 23 | \( 1 - 5.02T + 23T^{2} \) |
| 29 | \( 1 - 9.47T + 29T^{2} \) |
| 31 | \( 1 + 6.10T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 - 6.20T + 43T^{2} \) |
| 47 | \( 1 + 8.96T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 - 3.28T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 71 | \( 1 - 5.85T + 71T^{2} \) |
| 73 | \( 1 - 6.69T + 73T^{2} \) |
| 79 | \( 1 + 7.17T + 79T^{2} \) |
| 83 | \( 1 - 5.88T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−8.474057972012491180811833767081, −7.73469704505332859291091648342, −7.02787761202081094011747294927, −6.42121381314516434378007264808, −5.38410382044722473524700769559, −4.68842739258789153118761083195, −3.53642447442153771106529561847, −3.06615577347969661756512045203, −2.26389373410148696314742427204, −0.60424608790646362532744039729,
0.60424608790646362532744039729, 2.26389373410148696314742427204, 3.06615577347969661756512045203, 3.53642447442153771106529561847, 4.68842739258789153118761083195, 5.38410382044722473524700769559, 6.42121381314516434378007264808, 7.02787761202081094011747294927, 7.73469704505332859291091648342, 8.474057972012491180811833767081
