L(s) = 1 | − 0.879·3-s − 5-s − 1.22·7-s − 2.22·9-s − 1.71·11-s + 2·13-s + 0.879·15-s + 2.69·17-s + 8.12·19-s + 1.07·21-s + 3.75·23-s + 25-s + 4.59·27-s − 6.12·29-s + 7.92·31-s + 1.50·33-s + 1.22·35-s + 0.283·37-s − 1.75·39-s − 1.57·41-s − 43-s + 2.22·45-s + 2.69·47-s − 5.49·49-s − 2.36·51-s + 12.5·53-s + 1.71·55-s + ⋯ |
L(s) = 1 | − 0.507·3-s − 0.447·5-s − 0.463·7-s − 0.742·9-s − 0.517·11-s + 0.554·13-s + 0.227·15-s + 0.653·17-s + 1.86·19-s + 0.235·21-s + 0.783·23-s + 0.200·25-s + 0.884·27-s − 1.13·29-s + 1.42·31-s + 0.262·33-s + 0.207·35-s + 0.0465·37-s − 0.281·39-s − 0.245·41-s − 0.152·43-s + 0.331·45-s + 0.393·47-s − 0.785·49-s − 0.331·51-s + 1.72·53-s + 0.231·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.040698767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.040698767\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 0.879T + 3T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 - 8.12T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 + 6.12T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 - 0.283T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 47 | \( 1 - 2.69T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 0.0564T + 59T^{2} \) |
| 61 | \( 1 - 7.67T + 61T^{2} \) |
| 67 | \( 1 + 1.43T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 1.63T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 0.241T + 97T^{2} \) |
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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
−10.20763957824043044555067021260, −9.396618501028789779211770407429, −8.426480579230533563336040349645, −7.62186035567967295729740048033, −6.70533816374799649936861590195, −5.66423690983522174946307581327, −5.08894066228505235833413450958, −3.63942934501337755402899208732, −2.83397412114709563397324113785, −0.846635064070443714536925733391,
0.846635064070443714536925733391, 2.83397412114709563397324113785, 3.63942934501337755402899208732, 5.08894066228505235833413450958, 5.66423690983522174946307581327, 6.70533816374799649936861590195, 7.62186035567967295729740048033, 8.426480579230533563336040349645, 9.396618501028789779211770407429, 10.20763957824043044555067021260