L(s) = 1 | − 5-s − 7-s − 3·9-s − 2·11-s − 2·13-s + 3·17-s + 2·19-s − 23-s + 25-s + 7·29-s + 5·31-s + 35-s + 11·37-s + 41-s + 3·45-s − 6·49-s + 11·53-s + 2·55-s + 13·59-s − 8·61-s + 3·63-s + 2·65-s − 5·67-s − 5·71-s + 6·73-s + 2·77-s + 12·79-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s − 0.603·11-s − 0.554·13-s + 0.727·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 1.29·29-s + 0.898·31-s + 0.169·35-s + 1.80·37-s + 0.156·41-s + 0.447·45-s − 6/7·49-s + 1.51·53-s + 0.269·55-s + 1.69·59-s − 1.02·61-s + 0.377·63-s + 0.248·65-s − 0.610·67-s − 0.593·71-s + 0.702·73-s + 0.227·77-s + 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.190913599\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.190913599\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.287436056802008861665654849050, −8.261450062978304340130490972772, −7.88313741440253118524665483256, −6.89664930841244341712826498598, −6.00165457583941898386124574131, −5.23992404060255109399587286374, −4.32075270011969972226267491278, −3.14448167474562522721276000501, −2.55324698595565232433692561243, −0.72059586748507280140651813487,
0.72059586748507280140651813487, 2.55324698595565232433692561243, 3.14448167474562522721276000501, 4.32075270011969972226267491278, 5.23992404060255109399587286374, 6.00165457583941898386124574131, 6.89664930841244341712826498598, 7.88313741440253118524665483256, 8.261450062978304340130490972772, 9.287436056802008861665654849050