| L(s) = 1 | + 23.3·3-s + 25·5-s − 90.1·7-s + 301.·9-s + 121·11-s − 1.04e3·13-s + 583.·15-s − 541.·17-s − 1.75e3·19-s − 2.10e3·21-s − 2.63e3·23-s + 625·25-s + 1.36e3·27-s + 599.·29-s + 3.51e3·31-s + 2.82e3·33-s − 2.25e3·35-s + 739.·37-s − 2.43e4·39-s − 1.91e4·41-s − 1.34e4·43-s + 7.53e3·45-s + 1.04e4·47-s − 8.68e3·49-s − 1.26e4·51-s + 3.06e4·53-s + 3.02e3·55-s + ⋯ |
| L(s) = 1 | + 1.49·3-s + 0.447·5-s − 0.695·7-s + 1.24·9-s + 0.301·11-s − 1.71·13-s + 0.669·15-s − 0.454·17-s − 1.11·19-s − 1.04·21-s − 1.03·23-s + 0.200·25-s + 0.360·27-s + 0.132·29-s + 0.656·31-s + 0.451·33-s − 0.310·35-s + 0.0887·37-s − 2.56·39-s − 1.78·41-s − 1.10·43-s + 0.554·45-s + 0.693·47-s − 0.516·49-s − 0.679·51-s + 1.49·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 - 23.3T + 243T^{2} \) |
| 7 | \( 1 + 90.1T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 541.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.75e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.63e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 599.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 739.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.91e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.63e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.49e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.71e4T + 8.58e9T^{2} \) |
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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
−9.830490556210827637532298161215, −8.957494816287873596659115983645, −8.212638475816671049877841943896, −7.17784529987555340227272574504, −6.32709343225617743880522866344, −4.82720833600291134727184454668, −3.72074688374066094578967738530, −2.62556893828749544044026137111, −1.95142864434881258989482493602, 0,
1.95142864434881258989482493602, 2.62556893828749544044026137111, 3.72074688374066094578967738530, 4.82720833600291134727184454668, 6.32709343225617743880522866344, 7.17784529987555340227272574504, 8.212638475816671049877841943896, 8.957494816287873596659115983645, 9.830490556210827637532298161215
