| L(s) = 1 | + 0.757·3-s + 10.6·5-s − 22.1·7-s − 26.4·9-s − 39.3·11-s − 23.7·13-s + 8.07·15-s + 4.54·17-s + 155.·19-s − 16.7·21-s − 41.8·23-s − 11.4·25-s − 40.4·27-s − 29·29-s − 57.9·31-s − 29.7·33-s − 235.·35-s − 235.·37-s − 18.0·39-s − 175.·41-s + 402.·43-s − 281.·45-s + 227.·47-s + 147.·49-s + 3.44·51-s − 673.·53-s − 419.·55-s + ⋯ |
| L(s) = 1 | + 0.145·3-s + 0.953·5-s − 1.19·7-s − 0.978·9-s − 1.07·11-s − 0.507·13-s + 0.138·15-s + 0.0648·17-s + 1.87·19-s − 0.174·21-s − 0.379·23-s − 0.0914·25-s − 0.288·27-s − 0.185·29-s − 0.335·31-s − 0.157·33-s − 1.13·35-s − 1.04·37-s − 0.0739·39-s − 0.667·41-s + 1.42·43-s − 0.932·45-s + 0.706·47-s + 0.429·49-s + 0.00944·51-s − 1.74·53-s − 1.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.466373694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.466373694\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 0.757T + 27T^{2} \) |
| 5 | \( 1 - 10.6T + 125T^{2} \) |
| 7 | \( 1 + 22.1T + 343T^{2} \) |
| 11 | \( 1 + 39.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.8T + 1.21e4T^{2} \) |
| 31 | \( 1 + 57.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 175.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 402.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 227.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 673.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 800.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 222.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 524.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 515.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 358.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 829.T + 9.12e5T^{2} \) |
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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
−9.111354167251516207551521307130, −8.074222267481988160080804991160, −7.31687091415235579985720472760, −6.39841819151837979958539482796, −5.53498564000533182793210320369, −5.23181109589638870392600909325, −3.58856983184278774633463402859, −2.88202057417799768649202659400, −2.10526997096421772415782900806, −0.52912452322119245020990641007,
0.52912452322119245020990641007, 2.10526997096421772415782900806, 2.88202057417799768649202659400, 3.58856983184278774633463402859, 5.23181109589638870392600909325, 5.53498564000533182793210320369, 6.39841819151837979958539482796, 7.31687091415235579985720472760, 8.074222267481988160080804991160, 9.111354167251516207551521307130
