| L(s) = 1 | − 3.23·5-s + 7-s + 1.23·11-s + 13-s − 6.47·17-s + 2.47·19-s + 4.47·23-s + 5.47·25-s + 0.472·29-s − 3.23·35-s − 4.47·37-s − 0.763·41-s + 4·43-s + 5.70·47-s + 49-s − 10·53-s − 4.00·55-s − 4.76·59-s − 2.94·61-s − 3.23·65-s + 6.47·67-s − 7.70·71-s − 4.47·73-s + 1.23·77-s − 1.52·79-s − 7.23·83-s + 20.9·85-s + ⋯ |
| L(s) = 1 | − 1.44·5-s + 0.377·7-s + 0.372·11-s + 0.277·13-s − 1.56·17-s + 0.567·19-s + 0.932·23-s + 1.09·25-s + 0.0876·29-s − 0.546·35-s − 0.735·37-s − 0.119·41-s + 0.609·43-s + 0.832·47-s + 0.142·49-s − 1.37·53-s − 0.539·55-s − 0.620·59-s − 0.376·61-s − 0.401·65-s + 0.790·67-s − 0.914·71-s − 0.523·73-s + 0.140·77-s − 0.171·79-s − 0.794·83-s + 2.27·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.364828837452713215492537727074, −7.43444707441598201944660755023, −7.02124924228333819513922593357, −6.09632647639949358815922455380, −4.96828607233540435871214015072, −4.35053956777467132482456866170, −3.63907109143536877639932736802, −2.69521237164191251555075833936, −1.33521628267656554216921522928, 0,
1.33521628267656554216921522928, 2.69521237164191251555075833936, 3.63907109143536877639932736802, 4.35053956777467132482456866170, 4.96828607233540435871214015072, 6.09632647639949358815922455380, 7.02124924228333819513922593357, 7.43444707441598201944660755023, 8.364828837452713215492537727074
