| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s − 5.16·11-s + 3.16·13-s − 14-s + 16-s − 17-s − 1.16·19-s − 2·20-s + 5.16·22-s − 25-s − 3.16·26-s + 28-s + 9.48·29-s − 4.32·31-s − 32-s + 34-s − 2·35-s + 0.837·37-s + 1.16·38-s + 2·40-s − 5.16·44-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s − 1.55·11-s + 0.877·13-s − 0.267·14-s + 0.250·16-s − 0.242·17-s − 0.266·19-s − 0.447·20-s + 1.10·22-s − 0.200·25-s − 0.620·26-s + 0.188·28-s + 1.76·29-s − 0.776·31-s − 0.176·32-s + 0.171·34-s − 0.338·35-s + 0.137·37-s + 0.188·38-s + 0.316·40-s − 0.778·44-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8527766950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8527766950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 4.32T + 31T^{2} \) |
| 37 | \( 1 - 0.837T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.16T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.814871003047843248837491155669, −8.277655495258269910083404021016, −7.76336260280362373759422739578, −7.00217663412710727230969126010, −6.02243214184645859290596064016, −5.10535348378117278339014405379, −4.15274205477385405219873410899, −3.13044073003877362706928452598, −2.12226741697320113254378666783, −0.65756386023507038362389415289,
0.65756386023507038362389415289, 2.12226741697320113254378666783, 3.13044073003877362706928452598, 4.15274205477385405219873410899, 5.10535348378117278339014405379, 6.02243214184645859290596064016, 7.00217663412710727230969126010, 7.76336260280362373759422739578, 8.277655495258269910083404021016, 8.814871003047843248837491155669
