| L(s) = 1 | + 1.92·2-s + 1.71·4-s + 1.15·5-s + 2.35·7-s − 0.546·8-s + 2.23·10-s + 3.69·11-s + 1.22·13-s + 4.54·14-s − 4.48·16-s − 4.29·17-s + 6.29·19-s + 1.98·20-s + 7.12·22-s − 0.829·23-s − 3.65·25-s + 2.36·26-s + 4.04·28-s − 0.0257·29-s + 3.97·31-s − 7.55·32-s − 8.27·34-s + 2.73·35-s + 6.32·37-s + 12.1·38-s − 0.632·40-s − 7.01·41-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.858·4-s + 0.518·5-s + 0.891·7-s − 0.193·8-s + 0.706·10-s + 1.11·11-s + 0.340·13-s + 1.21·14-s − 1.12·16-s − 1.04·17-s + 1.44·19-s + 0.444·20-s + 1.51·22-s − 0.172·23-s − 0.731·25-s + 0.464·26-s + 0.764·28-s − 0.00478·29-s + 0.714·31-s − 1.33·32-s − 1.41·34-s + 0.461·35-s + 1.03·37-s + 1.96·38-s − 0.100·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.939484886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.939484886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 5 | \( 1 - 1.15T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + 0.829T + 23T^{2} \) |
| 29 | \( 1 + 0.0257T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + 7.01T + 41T^{2} \) |
| 43 | \( 1 - 9.18T + 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + 0.855T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 8.30T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 5.80T + 83T^{2} \) |
| 89 | \( 1 + 8.02T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−9.718819677029506296428165382887, −9.084953245672375698722295870552, −8.113926901103337776096905018227, −6.98419289746466424137745135400, −6.18581963985347061870207378873, −5.46675780510346304463342209346, −4.56243483563255917312567240158, −3.88553283016385834130192792947, −2.70398337430346528249372936907, −1.49548167744224560824752552620,
1.49548167744224560824752552620, 2.70398337430346528249372936907, 3.88553283016385834130192792947, 4.56243483563255917312567240158, 5.46675780510346304463342209346, 6.18581963985347061870207378873, 6.98419289746466424137745135400, 8.113926901103337776096905018227, 9.084953245672375698722295870552, 9.718819677029506296428165382887
