| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 12-s − 4·13-s + 14-s + 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s − 2·20-s + 21-s + 23-s + 24-s − 25-s + 4·26-s − 27-s − 28-s − 2·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.218·21-s + 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5722239630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5722239630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.41655651437687, −13.04891205815190, −12.33573807486696, −12.12905725755210, −11.48950051037212, −11.34159577786174, −10.68485184331613, −10.08333488912399, −9.761391930859082, −9.264050605921587, −8.695415080339143, −8.067393729663331, −7.566062464181197, −7.261411652339428, −6.751856796116963, −6.099697557320672, −5.654604622557322, −4.735289937417367, −4.652637255947972, −3.675020272585072, −3.250168292127314, −2.500340373798067, −1.868949925895422, −0.9304900097080833, −0.3353694525512261,
0.3353694525512261, 0.9304900097080833, 1.868949925895422, 2.500340373798067, 3.250168292127314, 3.675020272585072, 4.652637255947972, 4.735289937417367, 5.654604622557322, 6.099697557320672, 6.751856796116963, 7.261411652339428, 7.566062464181197, 8.067393729663331, 8.695415080339143, 9.264050605921587, 9.761391930859082, 10.08333488912399, 10.68485184331613, 11.34159577786174, 11.48950051037212, 12.12905725755210, 12.33573807486696, 13.04891205815190, 13.41655651437687
