| L(s) = 1 | − 2·4-s + 5-s − 7-s + 4·16-s + 3·17-s + 6·19-s − 2·20-s + 8·23-s − 4·25-s + 2·28-s + 29-s + 6·31-s − 35-s − 2·37-s − 6·41-s + 43-s − 3·47-s + 49-s − 12·53-s − 7·59-s + 12·61-s − 8·64-s − 11·67-s − 6·68-s − 8·71-s − 4·73-s − 12·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s + 16-s + 0.727·17-s + 1.37·19-s − 0.447·20-s + 1.66·23-s − 4/5·25-s + 0.377·28-s + 0.185·29-s + 1.07·31-s − 0.169·35-s − 0.328·37-s − 0.937·41-s + 0.152·43-s − 0.437·47-s + 1/7·49-s − 1.64·53-s − 0.911·59-s + 1.53·61-s − 64-s − 1.34·67-s − 0.727·68-s − 0.949·71-s − 0.468·73-s − 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.672654939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672654939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.18509951203728, −12.54014989435144, −12.04842133074260, −11.75072299314651, −11.04554330636122, −10.53341407297995, −9.967507048166699, −9.648153049754152, −9.404843111665107, −8.729373450157993, −8.376473763053181, −7.745989271143388, −7.341599461871784, −6.763148370518319, −6.106548261816406, −5.720529798445431, −5.027090712156039, −4.938435261279816, −4.163534747687304, −3.496478108045201, −3.093767372124982, −2.660250139223807, −1.470958370903904, −1.253890140657417, −0.3943194574524129,
0.3943194574524129, 1.253890140657417, 1.470958370903904, 2.660250139223807, 3.093767372124982, 3.496478108045201, 4.163534747687304, 4.938435261279816, 5.027090712156039, 5.720529798445431, 6.106548261816406, 6.763148370518319, 7.341599461871784, 7.745989271143388, 8.376473763053181, 8.729373450157993, 9.404843111665107, 9.648153049754152, 9.967507048166699, 10.53341407297995, 11.04554330636122, 11.75072299314651, 12.04842133074260, 12.54014989435144, 13.18509951203728
