| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 5·17-s + 7·19-s + 3·21-s + 8·23-s − 9·27-s + 3·29-s + 5·31-s + 37-s + 8·41-s − 10·43-s − 6·49-s − 15·51-s + 53-s − 21·57-s + 12·59-s + 5·61-s − 6·63-s − 4·67-s − 24·69-s + 7·71-s − 2·73-s − 4·79-s + 9·81-s − 9·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 1.21·17-s + 1.60·19-s + 0.654·21-s + 1.66·23-s − 1.73·27-s + 0.557·29-s + 0.898·31-s + 0.164·37-s + 1.24·41-s − 1.52·43-s − 6/7·49-s − 2.10·51-s + 0.137·53-s − 2.78·57-s + 1.56·59-s + 0.640·61-s − 0.755·63-s − 0.488·67-s − 2.88·69-s + 0.830·71-s − 0.234·73-s − 0.450·79-s + 81-s − 0.964·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.862095516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862095516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.08450105790162, −12.42078905763183, −12.13643666947867, −11.66115729219480, −11.22219700347123, −10.99231673345868, −10.10831165436761, −9.965515707393131, −9.659511456854094, −8.860830853744622, −8.306123234379178, −7.596911391960534, −7.162366887004558, −6.842180695933095, −6.163884235466138, −5.841875985194511, −5.226063339129440, −4.953360729402142, −4.477059378961815, −3.552033993258212, −3.228942650622742, −2.502759606858609, −1.436450885312322, −0.9549278445351826, −0.5792819257408772,
0.5792819257408772, 0.9549278445351826, 1.436450885312322, 2.502759606858609, 3.228942650622742, 3.552033993258212, 4.477059378961815, 4.953360729402142, 5.226063339129440, 5.841875985194511, 6.163884235466138, 6.842180695933095, 7.162366887004558, 7.596911391960534, 8.306123234379178, 8.860830853744622, 9.659511456854094, 9.965515707393131, 10.10831165436761, 10.99231673345868, 11.22219700347123, 11.66115729219480, 12.13643666947867, 12.42078905763183, 13.08450105790162
