L(s) = 1 | − i·3-s − 5-s − 7-s − 9-s − i·11-s − 13-s + i·15-s + i·17-s − i·19-s + i·21-s − 23-s + 25-s + i·27-s − i·31-s − 33-s + ⋯ |
L(s) = 1 | − i·3-s − 5-s − 7-s − 9-s − i·11-s − 13-s + i·15-s + i·17-s − i·19-s + i·21-s − 23-s + 25-s + i·27-s − i·31-s − 33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03733009779 - 0.3912410531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03733009779 - 0.3912410531i\) |
\(L(1)\) |
\(\approx\) |
\(0.5391112713 - 0.3042919672i\) |
\(L(1)\) |
\(\approx\) |
\(0.5391112713 - 0.3042919672i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 + iT \) |
| 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
−29.556113617516623894141016152245, −28.52095753225896007220757718666, −27.584213607499558887524707616854, −26.80492998762356080603862130668, −25.891893017225729278297468771661, −24.790527416825049408149010192253, −23.12613266161986415287110408250, −22.75230642086534898789815310603, −21.66408146334397881545216583011, −20.16754697113589318577854192605, −19.85337528992023898878098534267, −18.43489497682181113651312141054, −16.894133094382135384660853712788, −16.0701813867953237048504404120, −15.258601417960201156259534904783, −14.24600489399605487291621377095, −12.4947064476980806851363850591, −11.706261860876908796592832471973, −10.189151841839005927393591256, −9.53694293954025856428085568343, −8.05678606375511229368794689761, −6.77921202709610149935270001362, −5.07807663590806382556886208505, −4.005944547201673793635139517448, −2.84084847949748106858518580912,
0.35577537082832813059359961519, 2.55932305193514134932984533227, 3.83693375030246398657943032037, 5.76552553730440870922604223651, 6.91731204772854312514523580197, 7.91652383812084888996225059713, 9.052499128078792571058678567824, 10.75786821371137973335390480926, 11.94463216607519720318120543897, 12.7311753892512426175959059910, 13.82169363271607112607229622437, 15.13477001047866419986662386546, 16.30087174821338303288836270530, 17.31677024798406244971048448538, 18.76366779145127143528572761515, 19.41159586220681616452046285650, 20.0176049982896823717278034276, 21.89112199187841306323766723245, 22.74966161124073266197178575529, 23.951316273190174298772741533507, 24.32668463269994153183015937783, 25.82784782402170578870931769270, 26.55049256990118166865460511470, 27.90936275485432228407726252360, 28.90228116292172534744978175665