| L(s) = 1 | − 9·3-s − 53.5·5-s + 8.43·7-s + 81·9-s − 245.·11-s + 412.·13-s + 481.·15-s − 1.29e3·17-s + 379.·19-s − 75.9·21-s − 529·23-s − 258.·25-s − 729·27-s + 4.13e3·29-s + 4.92e3·31-s + 2.21e3·33-s − 451.·35-s + 1.28e3·37-s − 3.71e3·39-s + 5.05e3·41-s − 1.85e4·43-s − 4.33e3·45-s + 5.90e3·47-s − 1.67e4·49-s + 1.16e4·51-s + 3.57e4·53-s + 1.31e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.957·5-s + 0.0650·7-s + 0.333·9-s − 0.612·11-s + 0.676·13-s + 0.552·15-s − 1.08·17-s + 0.241·19-s − 0.0375·21-s − 0.208·23-s − 0.0826·25-s − 0.192·27-s + 0.913·29-s + 0.919·31-s + 0.353·33-s − 0.0623·35-s + 0.153·37-s − 0.390·39-s + 0.469·41-s − 1.53·43-s − 0.319·45-s + 0.389·47-s − 0.995·49-s + 0.625·51-s + 1.74·53-s + 0.586·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 53.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 8.43T + 1.68e4T^{2} \) |
| 11 | \( 1 + 245.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 412.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.29e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 379.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.05e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.85e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.90e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.24e5T + 8.58e9T^{2} \) |
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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
−8.467281061425495812872388987092, −8.004814129109752820874417697036, −6.96718245391218405149124809195, −6.27829840040500753732609111583, −5.19044935326742514897775139150, −4.39355557070091514216978161608, −3.53892640403886468651776681312, −2.32024705112153518963807318353, −0.938362224141067885364088866963, 0,
0.938362224141067885364088866963, 2.32024705112153518963807318353, 3.53892640403886468651776681312, 4.39355557070091514216978161608, 5.19044935326742514897775139150, 6.27829840040500753732609111583, 6.96718245391218405149124809195, 8.004814129109752820874417697036, 8.467281061425495812872388987092
