L(s) = 1 | − 3-s + 5-s − 2.21·7-s + 9-s + 1.26·11-s + 0.106·13-s − 15-s + 5.15·17-s − 4.68·19-s + 2.21·21-s + 8.51·23-s + 25-s − 27-s + 3.17·29-s − 5.36·31-s − 1.26·33-s − 2.21·35-s − 6.51·37-s − 0.106·39-s + 1.30·41-s − 3.73·43-s + 45-s − 4.77·47-s − 2.11·49-s − 5.15·51-s + 1.73·53-s + 1.26·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.835·7-s + 0.333·9-s + 0.380·11-s + 0.0294·13-s − 0.258·15-s + 1.25·17-s − 1.07·19-s + 0.482·21-s + 1.77·23-s + 0.200·25-s − 0.192·27-s + 0.588·29-s − 0.964·31-s − 0.219·33-s − 0.373·35-s − 1.07·37-s − 0.0169·39-s + 0.203·41-s − 0.570·43-s + 0.149·45-s − 0.697·47-s − 0.302·49-s − 0.722·51-s + 0.238·53-s + 0.170·55-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
\(L(1)\) |
\(\approx\) |
\(1.505437714\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.505437714\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 0.106T + 13T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 71 | \( 1 + 0.801T + 71T^{2} \) |
| 73 | \( 1 - 7.85T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 9.14T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.585751098811935695685477841892, −7.56924082607562299660645020763, −6.72141078231254622897892236088, −6.39479579989796767337749381299, −5.44141679521981970142684935390, −4.90472021721121244928417132551, −3.74049077664646844530647051370, −3.07684909585296802490105767859, −1.85919235632874187838730648086, −0.73230207267834792032508917628,
0.73230207267834792032508917628, 1.85919235632874187838730648086, 3.07684909585296802490105767859, 3.74049077664646844530647051370, 4.90472021721121244928417132551, 5.44141679521981970142684935390, 6.39479579989796767337749381299, 6.72141078231254622897892236088, 7.56924082607562299660645020763, 8.585751098811935695685477841892