| L(s) = 1 | − 2·5-s − 4·7-s − 4·11-s + 13-s − 2·17-s + 8·19-s − 25-s − 6·29-s + 4·31-s + 8·35-s − 2·37-s + 10·41-s − 4·43-s + 8·47-s + 9·49-s + 10·53-s + 8·55-s + 4·59-s − 2·61-s − 2·65-s + 16·67-s − 8·71-s + 2·73-s + 16·77-s − 8·79-s + 12·83-s + 4·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 1.20·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.35·35-s − 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s + 1.07·55-s + 0.520·59-s − 0.256·61-s − 0.248·65-s + 1.95·67-s − 0.949·71-s + 0.234·73-s + 1.82·77-s − 0.900·79-s + 1.31·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8374095724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8374095724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.345209736202926164014285850515, −8.395061304144126538425647052449, −7.52465381698855157679013184836, −7.08365389517628564377574744262, −5.97887006638506534112113716589, −5.29861144478199351873889204714, −4.06906259281866598276648207762, −3.34937914407959923888310141579, −2.53219499761269661738776398133, −0.58989713961815991805077071088,
0.58989713961815991805077071088, 2.53219499761269661738776398133, 3.34937914407959923888310141579, 4.06906259281866598276648207762, 5.29861144478199351873889204714, 5.97887006638506534112113716589, 7.08365389517628564377574744262, 7.52465381698855157679013184836, 8.395061304144126538425647052449, 9.345209736202926164014285850515
