L(s) = 1 | − 2·3-s − 2·4-s + 3·5-s − 7-s + 9-s + 3·11-s + 4·12-s − 4·13-s − 6·15-s + 4·16-s − 3·17-s + 19-s − 6·20-s + 2·21-s + 4·25-s + 4·27-s + 2·28-s + 6·29-s − 4·31-s − 6·33-s − 3·35-s − 2·36-s + 2·37-s + 8·39-s − 6·41-s − 43-s − 6·44-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s − 1.54·15-s + 16-s − 0.727·17-s + 0.229·19-s − 1.34·20-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 1.04·33-s − 0.507·35-s − 1/3·36-s + 0.328·37-s + 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4532532444\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4532532444\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−18.12139870347555292446209785145, −17.42643155985520821799903376961, −16.70224717326536251258127220715, −14.50114185013199863152077458510, −13.35149180731186594820148200230, −12.06296001644352266090310246809, −10.24320445923116880350231460909, −9.180168549070062598241568188033, −6.39084306102520193435942181609, −5.03912355415234199771433602492,
5.03912355415234199771433602492, 6.39084306102520193435942181609, 9.180168549070062598241568188033, 10.24320445923116880350231460909, 12.06296001644352266090310246809, 13.35149180731186594820148200230, 14.50114185013199863152077458510, 16.70224717326536251258127220715, 17.42643155985520821799903376961, 18.12139870347555292446209785145