| L(s) = 1 | + 2.30·3-s + 2.57·5-s + 7-s + 2.30·9-s − 1.35·11-s − 4.19·13-s + 5.92·15-s + 4.92·17-s + 19-s + 2.30·21-s + 6.19·23-s + 1.62·25-s − 1.60·27-s + 0.760·29-s + 9.46·31-s − 3.11·33-s + 2.57·35-s − 5.27·37-s − 9.66·39-s + 1.23·41-s − 9.91·43-s + 5.92·45-s − 5.77·47-s + 49-s + 11.3·51-s − 1.15·53-s − 3.48·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 1.15·5-s + 0.377·7-s + 0.767·9-s − 0.407·11-s − 1.16·13-s + 1.53·15-s + 1.19·17-s + 0.229·19-s + 0.502·21-s + 1.29·23-s + 0.324·25-s − 0.308·27-s + 0.141·29-s + 1.70·31-s − 0.542·33-s + 0.435·35-s − 0.867·37-s − 1.54·39-s + 0.193·41-s − 1.51·43-s + 0.883·45-s − 0.841·47-s + 0.142·49-s + 1.58·51-s − 0.158·53-s − 0.469·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.066429632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.066429632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 2.57T + 5T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 0.760T + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 9.91T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 + 9.02T + 73T^{2} \) |
| 79 | \( 1 + 0.707T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−9.867685372579972423882973789450, −9.129902162808659327454410557731, −8.260063285453490065016181044687, −7.62503930643274244981903873719, −6.66369850659263368980528602951, −5.45968424402866849925555058094, −4.77267575989166396803170459325, −3.24226605979583439501002342050, −2.59134385510800032433037633099, −1.54418085675601260196609098085,
1.54418085675601260196609098085, 2.59134385510800032433037633099, 3.24226605979583439501002342050, 4.77267575989166396803170459325, 5.45968424402866849925555058094, 6.66369850659263368980528602951, 7.62503930643274244981903873719, 8.260063285453490065016181044687, 9.129902162808659327454410557731, 9.867685372579972423882973789450
