| L(s) = 1 | + 2.41·3-s − 5-s − 1.41·7-s + 2.82·9-s − 11-s − 1.41·13-s − 2.41·15-s + 1.41·17-s + 19-s − 3.41·21-s − 4.65·23-s − 4·25-s − 0.414·27-s − 8.24·29-s − 3.24·31-s − 2.41·33-s + 1.41·35-s − 8.41·37-s − 3.41·39-s − 4·41-s + 6·43-s − 2.82·45-s + 6.48·47-s − 5·49-s + 3.41·51-s + 1.17·53-s + 55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s − 0.447·5-s − 0.534·7-s + 0.942·9-s − 0.301·11-s − 0.392·13-s − 0.623·15-s + 0.342·17-s + 0.229·19-s − 0.745·21-s − 0.971·23-s − 0.800·25-s − 0.0797·27-s − 1.53·29-s − 0.582·31-s − 0.420·33-s + 0.239·35-s − 1.38·37-s − 0.546·39-s − 0.624·41-s + 0.914·43-s − 0.421·45-s + 0.945·47-s − 0.714·49-s + 0.478·51-s + 0.160·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 7.24T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 - 9.72T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 - 0.485T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
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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
−8.232146217365321276452608871777, −7.59076214887811543858366997135, −7.14016426793018047486718000473, −5.98141571964991807090703427256, −5.18114895656635210261394260722, −3.85934893722556026115977078855, −3.63431240797480035424033975584, −2.59631003240840653835118506057, −1.81533305271407557602389132153, 0,
1.81533305271407557602389132153, 2.59631003240840653835118506057, 3.63431240797480035424033975584, 3.85934893722556026115977078855, 5.18114895656635210261394260722, 5.98141571964991807090703427256, 7.14016426793018047486718000473, 7.59076214887811543858366997135, 8.232146217365321276452608871777
