| L(s) = 1 | − 8·4-s − 37·7-s − 70·13-s + 64·16-s − 163·19-s + 296·28-s − 19·31-s − 433·37-s + 449·43-s + 1.02e3·49-s + 560·52-s + 719·61-s − 512·64-s − 880·67-s − 271·73-s + 1.30e3·76-s + 503·79-s + 2.59e3·91-s − 523·97-s − 19·103-s + 2.21e3·109-s − 2.36e3·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.99·7-s − 1.49·13-s + 16-s − 1.96·19-s + 1.99·28-s − 0.110·31-s − 1.92·37-s + 1.59·43-s + 2.99·49-s + 1.49·52-s + 1.50·61-s − 64-s − 1.60·67-s − 0.434·73-s + 1.96·76-s + 0.716·79-s + 2.98·91-s − 0.547·97-s − 0.0181·103-s + 1.94·109-s − 1.99·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3118855566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3118855566\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 37 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 19 T + p^{3} T^{2} \) |
| 37 | \( 1 + 433 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 449 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 719 T + p^{3} T^{2} \) |
| 67 | \( 1 + 880 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 271 T + p^{3} T^{2} \) |
| 79 | \( 1 - 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 523 T + p^{3} T^{2} \) |
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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
−10.05301910517370804996428942618, −9.254549739270288771210563106074, −8.664314040436831270509808234992, −7.37456800467077168454608130274, −6.53908112621475780826823398011, −5.60528463357822634209116223449, −4.45742616006974798436212380051, −3.56661797203057567929459651624, −2.44758715374932890799598401928, −0.30485927730869586333584222951,
0.30485927730869586333584222951, 2.44758715374932890799598401928, 3.56661797203057567929459651624, 4.45742616006974798436212380051, 5.60528463357822634209116223449, 6.53908112621475780826823398011, 7.37456800467077168454608130274, 8.664314040436831270509808234992, 9.254549739270288771210563106074, 10.05301910517370804996428942618
