| L(s) = 1 | − 1.49·2-s + 2.85·3-s + 0.230·4-s + 5-s − 4.26·6-s + 0.0973·7-s + 2.64·8-s + 5.16·9-s − 1.49·10-s − 11-s + 0.659·12-s + 7.01·13-s − 0.145·14-s + 2.85·15-s − 4.40·16-s − 1.04·17-s − 7.70·18-s + 2.98·19-s + 0.230·20-s + 0.278·21-s + 1.49·22-s + 0.0638·23-s + 7.54·24-s + 25-s − 10.4·26-s + 6.17·27-s + 0.0224·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 1.64·3-s + 0.115·4-s + 0.447·5-s − 1.74·6-s + 0.0367·7-s + 0.934·8-s + 1.72·9-s − 0.472·10-s − 0.301·11-s + 0.190·12-s + 1.94·13-s − 0.0388·14-s + 0.737·15-s − 1.10·16-s − 0.254·17-s − 1.81·18-s + 0.684·19-s + 0.0516·20-s + 0.0606·21-s + 0.318·22-s + 0.0133·23-s + 1.54·24-s + 0.200·25-s − 2.05·26-s + 1.18·27-s + 0.00424·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.401245273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401245273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 3 | \( 1 - 2.85T + 3T^{2} \) |
| 7 | \( 1 - 0.0973T + 7T^{2} \) |
| 13 | \( 1 - 7.01T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 23 | \( 1 - 0.0638T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 + 0.932T + 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 + 1.13T + 41T^{2} \) |
| 43 | \( 1 + 7.24T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 9.47T + 61T^{2} \) |
| 67 | \( 1 - 1.84T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 79 | \( 1 - 5.60T + 79T^{2} \) |
| 83 | \( 1 + 7.12T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 0.429T + 97T^{2} \) |
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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
−8.621058176118451542080746896916, −8.052828884612227013909475603881, −7.36773787920047854232849243554, −6.56630864666295794060365458769, −5.52508031480038767292202018600, −4.39490650184797675641122533931, −3.66096917357354679167295623774, −2.80640571727040826713322970771, −1.80931119542492191562317489074, −1.06392084027032631914694677316,
1.06392084027032631914694677316, 1.80931119542492191562317489074, 2.80640571727040826713322970771, 3.66096917357354679167295623774, 4.39490650184797675641122533931, 5.52508031480038767292202018600, 6.56630864666295794060365458769, 7.36773787920047854232849243554, 8.052828884612227013909475603881, 8.621058176118451542080746896916
