| L(s) = 1 | + 2.44·5-s + 1.03·7-s + 1.26·11-s + 5.27·13-s − 3.46·17-s − 1.26·19-s + 6.69·23-s + 0.999·25-s − 2.44·29-s + 5.65·31-s + 2.53·35-s − 0.378·37-s + 6.92·41-s − 8.19·43-s + 9.79·47-s − 5.92·49-s − 6.03·53-s + 3.10·55-s + 10.7·59-s + 0.378·61-s + 12.9·65-s + 4.19·67-s − 6.69·71-s − 9.46·73-s + 1.31·77-s + 15.4·79-s + 8.19·83-s + ⋯ |
| L(s) = 1 | + 1.09·5-s + 0.391·7-s + 0.382·11-s + 1.46·13-s − 0.840·17-s − 0.290·19-s + 1.39·23-s + 0.199·25-s − 0.454·29-s + 1.01·31-s + 0.428·35-s − 0.0622·37-s + 1.08·41-s − 1.24·43-s + 1.42·47-s − 0.846·49-s − 0.829·53-s + 0.418·55-s + 1.39·59-s + 0.0485·61-s + 1.60·65-s + 0.512·67-s − 0.794·71-s − 1.10·73-s + 0.149·77-s + 1.73·79-s + 0.899·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.253585072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.253585072\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 5.27T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 6.69T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 0.378T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 6.03T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 0.378T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 + 9.46T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−7.78266637422564464775332830345, −6.77516719406355602615103496374, −6.39697398026303304484508453401, −5.74010148591775733147848684921, −5.01903082420592550042201311864, −4.26112353198806899539648154890, −3.44304185538197240726352333669, −2.49425333108933650414003467380, −1.70712505690174804079330533861, −0.929223168791297267810133906094,
0.929223168791297267810133906094, 1.70712505690174804079330533861, 2.49425333108933650414003467380, 3.44304185538197240726352333669, 4.26112353198806899539648154890, 5.01903082420592550042201311864, 5.74010148591775733147848684921, 6.39697398026303304484508453401, 6.77516719406355602615103496374, 7.78266637422564464775332830345
