| L(s) = 1 | + 7-s − 4·11-s − 4·13-s + 2·17-s + 6·19-s + 8·23-s − 5·25-s − 2·29-s + 4·31-s + 10·37-s + 10·41-s − 4·43-s + 4·47-s + 49-s + 2·53-s + 10·59-s − 8·61-s + 8·67-s − 6·73-s − 4·77-s + 16·79-s + 2·83-s − 18·89-s − 4·91-s − 2·97-s + 4·103-s − 16·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s − 1.10·13-s + 0.485·17-s + 1.37·19-s + 1.66·23-s − 25-s − 0.371·29-s + 0.718·31-s + 1.64·37-s + 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.274·53-s + 1.30·59-s − 1.02·61-s + 0.977·67-s − 0.702·73-s − 0.455·77-s + 1.80·79-s + 0.219·83-s − 1.90·89-s − 0.419·91-s − 0.203·97-s + 0.394·103-s − 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677594917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677594917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.374383382686211568270374196986, −8.142737265603496815088661734936, −7.64230674264070832489473958476, −7.03619200467257105345639081242, −5.72643285643591914021095909045, −5.20704742703837539203927825545, −4.38921941756740989472338293857, −3.09009811816266066443914661488, −2.38160133368819946421648201021, −0.867770609357001154731944493797,
0.867770609357001154731944493797, 2.38160133368819946421648201021, 3.09009811816266066443914661488, 4.38921941756740989472338293857, 5.20704742703837539203927825545, 5.72643285643591914021095909045, 7.03619200467257105345639081242, 7.64230674264070832489473958476, 8.142737265603496815088661734936, 9.374383382686211568270374196986
