L(s) = 1 | − 7-s − i·11-s − i·13-s + 17-s − i·19-s + 23-s − i·29-s − 31-s + i·37-s + 41-s − i·43-s − 47-s + 49-s − i·53-s − i·59-s + ⋯ |
L(s) = 1 | − 7-s − i·11-s − i·13-s + 17-s − i·19-s + 23-s − i·29-s − 31-s + i·37-s + 41-s − i·43-s − 47-s + 49-s − i·53-s − i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8179913775 - 0.5465643644i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8179913775 - 0.5465643644i\) |
\(L(1)\) |
\(\approx\) |
\(0.9255553623 - 0.2068141687i\) |
\(L(1)\) |
\(\approx\) |
\(0.9255553623 - 0.2068141687i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 \) |
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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
−26.15190749765898726681574360401, −25.57397055034247572071992023905, −24.65985991232625457672635792814, −23.269205593597592849126078812468, −22.94778824093387598606277580794, −21.717512542393328459808081755026, −20.84517494035208035922719620924, −19.77820810877779868925675222714, −18.96454883992551183042288324927, −18.072175547236951136101345203296, −16.72994815093635294046256458765, −16.251685411371243413094899115033, −14.94183823268132981244966016346, −14.14817469067739343819335936206, −12.79506717404215774686839534226, −12.280424435601826505470368736449, −10.914312953723082174110329258038, −9.78023560555355267432743021397, −9.12411298931315809830848283879, −7.59070552961348869572795779873, −6.72917205762816271041628317393, −5.56172289228254004552301182437, −4.21582448339759127435915467088, −3.08835759373921316614357212660, −1.60847144595022416460358667338,
0.739849064957020356426768335863, 2.78557517359073033875421664225, 3.57518561733692363749664924493, 5.231993443398095803011504825059, 6.16549271699470535027648507755, 7.327438240311009893739750044791, 8.50568310426989267554989362471, 9.56160650572696755496255185412, 10.54506955275824459249650229629, 11.602186668077718863957088870425, 12.86096671933584119990289189374, 13.43694810752523968771364938888, 14.73186433173428841512812842450, 15.733970600914965541872646642716, 16.56075638110085335563334464985, 17.51762032108246927399050942233, 18.78156681358589885767477273910, 19.36960133500664622071692062355, 20.420248273828803752057179682655, 21.475318299398398128156326144500, 22.377658215580481820578206361346, 23.16316984729890701729851397392, 24.194305218230777763329711252246, 25.21069262145087570384247973178, 25.9311018812890220116832208078