L(s) = 1 | − 2-s − 4-s − 5-s − 7-s + 3·8-s + 10-s − 3·11-s + 7·13-s + 14-s − 16-s + 8·19-s + 20-s + 3·22-s + 4·23-s + 25-s − 7·26-s + 28-s + 2·29-s + 5·31-s − 5·32-s + 35-s − 2·37-s − 8·38-s − 3·40-s + 3·41-s + 4·43-s + 3·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.316·10-s − 0.904·11-s + 1.94·13-s + 0.267·14-s − 1/4·16-s + 1.83·19-s + 0.223·20-s + 0.639·22-s + 0.834·23-s + 1/5·25-s − 1.37·26-s + 0.188·28-s + 0.371·29-s + 0.898·31-s − 0.883·32-s + 0.169·35-s − 0.328·37-s − 1.29·38-s − 0.474·40-s + 0.468·41-s + 0.609·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.471827748\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.471827748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.79222883922147, −13.45627551010664, −12.91214013609324, −12.48153794407369, −11.79239566132857, −11.17786298336480, −10.85405780493003, −10.41910168727345, −9.764305275494274, −9.293577668372230, −8.952744061121361, −8.279016667304553, −7.809853960919904, −7.672362993055232, −6.720499875582271, −6.337294230700521, −5.559728538972132, −5.032972172677249, −4.579863426094250, −3.735235308237946, −3.330356185392151, −2.813487908415975, −1.678677858506165, −1.030439896090082, −0.5542969656899008,
0.5542969656899008, 1.030439896090082, 1.678677858506165, 2.813487908415975, 3.330356185392151, 3.735235308237946, 4.579863426094250, 5.032972172677249, 5.559728538972132, 6.337294230700521, 6.720499875582271, 7.672362993055232, 7.809853960919904, 8.279016667304553, 8.952744061121361, 9.293577668372230, 9.764305275494274, 10.41910168727345, 10.85405780493003, 11.17786298336480, 11.79239566132857, 12.48153794407369, 12.91214013609324, 13.45627551010664, 13.79222883922147