| L(s) = 1 | + 3-s + 5·7-s − 2·9-s − 11-s − 2·13-s − 3·17-s + 7·19-s + 5·21-s − 6·23-s − 5·27-s − 3·29-s + 7·31-s − 33-s + 7·37-s − 2·39-s + 6·41-s + 8·43-s + 6·47-s + 18·49-s − 3·51-s + 3·53-s + 7·57-s + 6·59-s − 61-s − 10·63-s + 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s − 2/3·9-s − 0.301·11-s − 0.554·13-s − 0.727·17-s + 1.60·19-s + 1.09·21-s − 1.25·23-s − 0.962·27-s − 0.557·29-s + 1.25·31-s − 0.174·33-s + 1.15·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.875·47-s + 18/7·49-s − 0.420·51-s + 0.412·53-s + 0.927·57-s + 0.781·59-s − 0.128·61-s − 1.25·63-s + 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.782440118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.782440118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.144629570314104299064609173266, −7.84647397900658161148191423110, −7.25857771125828668817246324494, −5.95953114710311995100355399382, −5.38197724822170240785431134634, −4.60521843157884328377810567555, −3.90352916818837526890905076594, −2.62397526533186342676974067130, −2.17432538238980105915986977099, −0.933901831255390711036545694034,
0.933901831255390711036545694034, 2.17432538238980105915986977099, 2.62397526533186342676974067130, 3.90352916818837526890905076594, 4.60521843157884328377810567555, 5.38197724822170240785431134634, 5.95953114710311995100355399382, 7.25857771125828668817246324494, 7.84647397900658161148191423110, 8.144629570314104299064609173266
