| L(s) = 1 | + 3·3-s − 2·5-s + 3·7-s + 6·9-s − 6·11-s + 5·13-s − 6·15-s + 6·17-s + 19-s + 9·21-s − 5·23-s − 25-s + 9·27-s + 2·29-s + 5·31-s − 18·33-s − 6·35-s + 2·37-s + 15·39-s + 10·41-s + 43-s − 12·45-s + 47-s + 2·49-s + 18·51-s − 6·53-s + 12·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s + 1.13·7-s + 2·9-s − 1.80·11-s + 1.38·13-s − 1.54·15-s + 1.45·17-s + 0.229·19-s + 1.96·21-s − 1.04·23-s − 1/5·25-s + 1.73·27-s + 0.371·29-s + 0.898·31-s − 3.13·33-s − 1.01·35-s + 0.328·37-s + 2.40·39-s + 1.56·41-s + 0.152·43-s − 1.78·45-s + 0.145·47-s + 2/7·49-s + 2.52·51-s − 0.824·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.773700700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.773700700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.118919760394265003862035406240, −7.76885555096563642301071969951, −7.61929517576046001402227529231, −6.15216908346320224042813166730, −5.20463019364776179827510463711, −4.34058525248639309681056856894, −3.66335294918981601416472585770, −2.95473872018033772487398854687, −2.13153037242563482838329390732, −1.06101048667711343840922297454,
1.06101048667711343840922297454, 2.13153037242563482838329390732, 2.95473872018033772487398854687, 3.66335294918981601416472585770, 4.34058525248639309681056856894, 5.20463019364776179827510463711, 6.15216908346320224042813166730, 7.61929517576046001402227529231, 7.76885555096563642301071969951, 8.118919760394265003862035406240
