| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s + 15-s + 17-s − 19-s + 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s + 35-s + 37-s − 41-s + 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s − 57-s − 59-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s + 15-s + 17-s − 19-s + 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s + 35-s + 37-s − 41-s + 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s − 57-s − 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.026961367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.026961367\) |
| \(L(1)\) |
\(\approx\) |
\(1.848351028\) |
| \(L(1)\) |
\(\approx\) |
\(1.848351028\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.84701989611595979887838415498, −28.44564835155039314267509579535, −27.34683579446599695703274000034, −26.179035426298324744753156035, −25.5136042256090969234284742691, −24.49165996135023118833477980327, −23.58106383718192372947967658110, −21.803411450417782748075892772944, −21.041348030693650768374336146779, −20.41963646556510260687141719149, −18.86123399616252282130699874817, −18.093458630888906728726652285, −16.88702495720588850573967598507, −15.38870157540188232768568885581, −14.39948108547307587363533898108, −13.60803938070336577775910494778, −12.498785326255633041266279140494, −10.70081274577193705310219431143, −9.76060647617053544131734412883, −8.46879361640387085847337206627, −7.56397652238752042157472116127, −5.827336206221570025657218411822, −4.4680506144893698448488608055, −2.70191826042558408591819428205, −1.62049583024508401057869089681,
1.62049583024508401057869089681, 2.70191826042558408591819428205, 4.4680506144893698448488608055, 5.827336206221570025657218411822, 7.56397652238752042157472116127, 8.46879361640387085847337206627, 9.76060647617053544131734412883, 10.70081274577193705310219431143, 12.498785326255633041266279140494, 13.60803938070336577775910494778, 14.39948108547307587363533898108, 15.38870157540188232768568885581, 16.88702495720588850573967598507, 18.093458630888906728726652285, 18.86123399616252282130699874817, 20.41963646556510260687141719149, 21.041348030693650768374336146779, 21.803411450417782748075892772944, 23.58106383718192372947967658110, 24.49165996135023118833477980327, 25.5136042256090969234284742691, 26.179035426298324744753156035, 27.34683579446599695703274000034, 28.44564835155039314267509579535, 29.84701989611595979887838415498
