| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 13-s − 17-s − 19-s + 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 37-s − 39-s + 41-s + 43-s + 47-s + 49-s − 51-s − 53-s − 57-s − 59-s + 61-s + 63-s + 67-s + 69-s + ⋯ |
| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 13-s − 17-s − 19-s + 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 37-s − 39-s + 41-s + 43-s + 47-s + 49-s − 51-s − 53-s − 57-s − 59-s + 61-s + 63-s + 67-s + 69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.679671111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679671111\) |
| \(L(1)\) |
\(\approx\) |
\(1.404962946\) |
| \(L(1)\) |
\(\approx\) |
\(1.404962946\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 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
−39.80481063438160399687206257088, −38.36118664682136497386941158032, −37.05093651805288071763400427155, −36.351014501254046329911488825377, −34.51853083486020412867075384299, −33.16200675697415676234365610618, −31.629850769137191816888066837853, −30.813432186697565045945988246015, −29.31563646294754198450981764783, −27.42941077800643637202735468112, −26.378159592740234285714437101144, −24.90661039953541278981646848032, −23.786167100158956031134321134801, −21.62220640648137065696932558496, −20.54210870445473529316857194361, −19.12025280279851150342337755208, −17.60073479127063360348826513537, −15.49353496124302680886517579200, −14.33616900750051547377242364324, −12.80240081086662892203384420634, −10.66330073495362225432062668787, −8.80452742454490020235868190318, −7.42910977458417845335920059741, −4.67550774984207957491488368557, −2.35893499408665604861501236977,
2.35893499408665604861501236977, 4.67550774984207957491488368557, 7.42910977458417845335920059741, 8.80452742454490020235868190318, 10.66330073495362225432062668787, 12.80240081086662892203384420634, 14.33616900750051547377242364324, 15.49353496124302680886517579200, 17.60073479127063360348826513537, 19.12025280279851150342337755208, 20.54210870445473529316857194361, 21.62220640648137065696932558496, 23.786167100158956031134321134801, 24.90661039953541278981646848032, 26.378159592740234285714437101144, 27.42941077800643637202735468112, 29.31563646294754198450981764783, 30.813432186697565045945988246015, 31.629850769137191816888066837853, 33.16200675697415676234365610618, 34.51853083486020412867075384299, 36.351014501254046329911488825377, 37.05093651805288071763400427155, 38.36118664682136497386941158032, 39.80481063438160399687206257088
