L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 3·9-s − 2·10-s − 4·11-s + 14-s + 16-s + 6·17-s + 3·18-s − 4·19-s + 2·20-s + 4·22-s + 23-s − 25-s − 28-s − 2·31-s − 32-s − 6·34-s − 2·35-s − 3·36-s + 6·37-s + 4·38-s − 2·40-s − 8·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.632·10-s − 1.20·11-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.208·23-s − 1/5·25-s − 0.188·28-s − 0.359·31-s − 0.176·32-s − 1.02·34-s − 0.338·35-s − 1/2·36-s + 0.986·37-s + 0.648·38-s − 0.316·40-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.131676507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131676507\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.83831117539889, −12.37495623774608, −11.72904301678785, −11.25922852514550, −10.96217647704787, −10.15139698501665, −10.11026120843751, −9.638537555183728, −9.182080785283972, −8.378514207638677, −8.304921405487035, −7.816201803094200, −7.179945161466013, −6.592859233195547, −6.176248062578340, −5.668608891343092, −5.289443822421728, −4.896969888323525, −3.866812001670807, −3.346679697505599, −2.835417893311676, −2.270958501262758, −1.912969829933270, −1.006480640550527, −0.3520749608189681,
0.3520749608189681, 1.006480640550527, 1.912969829933270, 2.270958501262758, 2.835417893311676, 3.346679697505599, 3.866812001670807, 4.896969888323525, 5.289443822421728, 5.668608891343092, 6.176248062578340, 6.592859233195547, 7.179945161466013, 7.816201803094200, 8.304921405487035, 8.378514207638677, 9.182080785283972, 9.638537555183728, 10.11026120843751, 10.15139698501665, 10.96217647704787, 11.25922852514550, 11.72904301678785, 12.37495623774608, 12.83831117539889