| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 11-s − 13-s + 14-s + 16-s − 17-s − 19-s + 22-s − 26-s + 28-s − 29-s + 31-s + 32-s − 34-s + 37-s − 38-s − 41-s + 43-s + 44-s + 47-s + 49-s − 52-s − 53-s + 56-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 11-s − 13-s + 14-s + 16-s − 17-s − 19-s + 22-s − 26-s + 28-s − 29-s + 31-s + 32-s − 34-s + 37-s − 38-s − 41-s + 43-s + 44-s + 47-s + 49-s − 52-s − 53-s + 56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.647033043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.647033043\) |
| \(L(1)\) |
\(\approx\) |
\(2.048451238\) |
| \(L(1)\) |
\(\approx\) |
\(2.048451238\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 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
−24.573860747972975949961128563660, −24.12229638792361441694949823914, −23.09868186267186723530025195265, −22.09237793518191652404389529983, −21.62197442483613512533109720360, −20.52905166388584956981516921061, −19.85959126149502736739013212363, −18.87690913221773090408387551491, −17.34974336446995004777324059840, −16.97408224416657874398166297865, −15.55781795967104477990847971306, −14.79794429642898695709941567962, −14.18639499809584748974370804323, −13.134293323070897607567231809312, −12.1186801057505446029987504624, −11.39570364563965566568597312445, −10.52347298078689620329078000580, −9.1347889008394234458891472860, −7.906220746044566267388142377329, −6.923150291588460431060781545341, −5.94237793105117534598120693087, −4.69029767487662773866993936497, −4.14886097920110186077870232560, −2.59591001066290431223861663725, −1.61409372396333121851805846463,
1.61409372396333121851805846463, 2.59591001066290431223861663725, 4.14886097920110186077870232560, 4.69029767487662773866993936497, 5.94237793105117534598120693087, 6.923150291588460431060781545341, 7.906220746044566267388142377329, 9.1347889008394234458891472860, 10.52347298078689620329078000580, 11.39570364563965566568597312445, 12.1186801057505446029987504624, 13.134293323070897607567231809312, 14.18639499809584748974370804323, 14.79794429642898695709941567962, 15.55781795967104477990847971306, 16.97408224416657874398166297865, 17.34974336446995004777324059840, 18.87690913221773090408387551491, 19.85959126149502736739013212363, 20.52905166388584956981516921061, 21.62197442483613512533109720360, 22.09237793518191652404389529983, 23.09868186267186723530025195265, 24.12229638792361441694949823914, 24.573860747972975949961128563660
