L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 3·17-s − 18-s − 6·19-s − 20-s − 21-s + 2·22-s − 4·23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3314127656\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3314127656\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.38371648740205, −12.06311571429019, −11.34730065964795, −11.07154842893683, −10.87410987253637, −10.18783298777622, −9.804449966773625, −9.472007604124303, −8.764383824244490, −8.322207886424493, −7.893191516937633, −7.550461102768171, −7.126382538792573, −6.469941845402739, −5.999986079809656, −5.652164258593846, −4.999316170802739, −4.527445324565045, −3.974734143911100, −3.479399454427646, −2.716030876423363, −2.170437759626321, −1.677244461950200, −0.9451641588840425, −0.2001430865952678,
0.2001430865952678, 0.9451641588840425, 1.677244461950200, 2.170437759626321, 2.716030876423363, 3.479399454427646, 3.974734143911100, 4.527445324565045, 4.999316170802739, 5.652164258593846, 5.999986079809656, 6.469941845402739, 7.126382538792573, 7.550461102768171, 7.893191516937633, 8.322207886424493, 8.764383824244490, 9.472007604124303, 9.804449966773625, 10.18783298777622, 10.87410987253637, 11.07154842893683, 11.34730065964795, 12.06311571429019, 12.38371648740205