| L(s) = 1 | + 3.29·5-s + 3.59·7-s − 11-s + 13-s + 8.12·17-s + 3.59·19-s + 3.74·23-s + 5.83·25-s − 3.69·29-s + 5.29·31-s + 11.8·35-s − 9.82·37-s − 8.08·41-s + 2.94·43-s − 8.73·47-s + 5.90·49-s + 8.58·53-s − 3.29·55-s − 6.58·59-s + 7.24·61-s + 3.29·65-s − 4.38·67-s − 9.33·71-s − 2.99·73-s − 3.59·77-s + 13.5·79-s + 4.09·83-s + ⋯ |
| L(s) = 1 | + 1.47·5-s + 1.35·7-s − 0.301·11-s + 0.277·13-s + 1.97·17-s + 0.824·19-s + 0.781·23-s + 1.16·25-s − 0.686·29-s + 0.950·31-s + 1.99·35-s − 1.61·37-s − 1.26·41-s + 0.448·43-s − 1.27·47-s + 0.844·49-s + 1.17·53-s − 0.443·55-s − 0.857·59-s + 0.927·61-s + 0.408·65-s − 0.535·67-s − 1.10·71-s − 0.350·73-s − 0.409·77-s + 1.52·79-s + 0.449·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.606135718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606135718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 17 | \( 1 - 8.12T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 + 8.08T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + 2.99T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.201744555200105810478141204888, −7.57700219329440861246231874274, −6.78487655905848869254044627251, −5.81799415963120179805312171212, −5.27766729962900904412882771807, −4.94293771987342701951827792434, −3.59946892654081484454120268205, −2.75728230803555524570948770966, −1.67204742655987855502480092122, −1.21559566493213847000174782306,
1.21559566493213847000174782306, 1.67204742655987855502480092122, 2.75728230803555524570948770966, 3.59946892654081484454120268205, 4.94293771987342701951827792434, 5.27766729962900904412882771807, 5.81799415963120179805312171212, 6.78487655905848869254044627251, 7.57700219329440861246231874274, 8.201744555200105810478141204888
