L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s − 2·7-s − 2·9-s + 2·10-s − 11-s − 2·12-s − 4·13-s − 4·14-s − 15-s − 4·16-s − 2·17-s − 4·18-s + 2·20-s + 2·21-s − 2·22-s + 23-s − 4·25-s − 8·26-s + 5·27-s − 4·28-s − 2·30-s − 7·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 1.10·13-s − 1.06·14-s − 0.258·15-s − 16-s − 0.485·17-s − 0.942·18-s + 0.447·20-s + 0.436·21-s − 0.426·22-s + 0.208·23-s − 4/5·25-s − 1.56·26-s + 0.962·27-s − 0.755·28-s − 0.365·30-s − 1.25·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38291 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38291 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3798434933\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3798434933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 59 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.71217477830291, −14.24775689029730, −13.77355401309867, −13.27019990544135, −12.63509788288163, −12.55067048496360, −11.74438948441375, −11.44552864958581, −10.80148897841731, −10.19855754357907, −9.552668906443518, −9.142707531607480, −8.430830990454663, −7.601376385687528, −6.905870736094264, −6.521645931517473, −5.761081129215114, −5.637918074886440, −4.828482711056518, −4.539379309149662, −3.489538831438976, −3.144377853174592, −2.416749976871860, −1.768860897655281, −0.1679845147825816,
0.1679845147825816, 1.768860897655281, 2.416749976871860, 3.144377853174592, 3.489538831438976, 4.539379309149662, 4.828482711056518, 5.637918074886440, 5.761081129215114, 6.521645931517473, 6.905870736094264, 7.601376385687528, 8.430830990454663, 9.142707531607480, 9.552668906443518, 10.19855754357907, 10.80148897841731, 11.44552864958581, 11.74438948441375, 12.55067048496360, 12.63509788288163, 13.27019990544135, 13.77355401309867, 14.24775689029730, 14.71217477830291