| L(s) = 1 | + 3.73·5-s − 3·9-s − 7.92·17-s + 8.92·25-s − 8.46·29-s − 7.73·37-s + 12.6·41-s − 11.1·45-s − 7·49-s − 10.4·53-s − 5.39·61-s + 2.80·73-s + 9·81-s − 29.5·85-s − 16·89-s − 8·97-s + 16.3·101-s − 20·109-s − 20.8·113-s + ⋯ |
| L(s) = 1 | + 1.66·5-s − 9-s − 1.92·17-s + 1.78·25-s − 1.57·29-s − 1.27·37-s + 1.97·41-s − 1.66·45-s − 49-s − 1.43·53-s − 0.690·61-s + 0.328·73-s + 81-s − 3.20·85-s − 1.69·89-s − 0.812·97-s + 1.62·101-s − 1.91·109-s − 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.88695849220262539095944527407, −6.85817179297181460565134176588, −6.31773250963612643999737295540, −5.69358746789878464183692862362, −5.11102231779794664847890717306, −4.18937653794930555689353306568, −3.01471572868245050908364521695, −2.28800704006449534940069097623, −1.62412847335338390326292005473, 0,
1.62412847335338390326292005473, 2.28800704006449534940069097623, 3.01471572868245050908364521695, 4.18937653794930555689353306568, 5.11102231779794664847890717306, 5.69358746789878464183692862362, 6.31773250963612643999737295540, 6.85817179297181460565134176588, 7.88695849220262539095944527407
