L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s + i·5-s + (−0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s + i·9-s + 10-s + (0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s + (−0.707 + 0.707i)17-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s + i·5-s + (−0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s + i·9-s + 10-s + (0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s + (−0.707 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0750 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0750 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09395277458 + 0.08714719745i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09395277458 + 0.08714719745i\) |
\(L(1)\) |
\(\approx\) |
\(0.4739978863 - 0.2394774269i\) |
\(L(1)\) |
\(\approx\) |
\(0.4739978863 - 0.2394774269i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (-0.707 + 0.707i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.707 - 0.707i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 + (-0.707 + 0.707i)T \) |
| 71 | \( 1 + (-0.707 - 0.707i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.707 - 0.707i)T \) |
| 97 | \( 1 + (0.707 - 0.707i)T \) |
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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
−34.39810981462501564026646039873, −33.16330147620147702275437106894, −32.17419639506517574962263794686, −31.61933008831481666931620195937, −29.249349961012518161784974315059, −28.16924411126427134359552885275, −27.28763717871179903822098605444, −26.02678574161552258090082958147, −24.63682639375904715474058414749, −23.7663102726138546898729660335, −22.253795714957897503343132936426, −21.62484222974837999552395700611, −19.64554667342186736974371034334, −17.98251838472865663579967117498, −16.61277469240706781053459800711, −16.19038985902923810996035415673, −14.82056026185019933683452993693, −13.06961312221571186073633317434, −11.75287133707356890447172117554, −9.52782823977403823828418359779, −8.8616148962070003464452704085, −6.59981178430745980042527556488, −5.40016439625234311682079426459, −4.13736276530136041190886177490, −0.08542869317513882108231456845,
2.097425253162804008824615751238, 3.99357600855204847494378128355, 6.15126891806959372974453490573, 7.59871391270992737450569187906, 9.93559418919097195280507491578, 10.85143185278566022490279681615, 12.21603576801177467501379759982, 13.27163137496520295514919293084, 14.67983652534658696874753809876, 17.01750906974896711592499764038, 17.9381315226362688458747659629, 19.18931125850204004357254870598, 19.99717153676378825066373277773, 22.09426271925881593268959272572, 22.59106695930226519820226732083, 23.65320241157785521100136790211, 25.54478715866147549208346868513, 26.894340073993925138166472931383, 28.0731710716963189718022623001, 29.39632484702879269400601137994, 29.96131056439772395835896267596, 30.84605377551410170513340093805, 32.49737695645042055895998251788, 33.84032236871474399933911183494, 35.25626539498504423458854651876