L(s) = 1 | + (−0.921 − 0.389i)2-s + (−0.809 − 0.587i)3-s + (0.696 + 0.717i)4-s + (−0.254 − 0.967i)5-s + (0.516 + 0.856i)6-s + (0.941 − 0.336i)7-s + (−0.362 − 0.931i)8-s + (0.309 + 0.951i)9-s + (−0.142 + 0.989i)10-s + (−0.142 − 0.989i)12-s + (0.897 − 0.441i)13-s + (−0.998 − 0.0570i)14-s + (−0.362 + 0.931i)15-s + (−0.0285 + 0.999i)16-s + (−0.736 − 0.676i)17-s + (0.0855 − 0.996i)18-s + ⋯ |
L(s) = 1 | + (−0.921 − 0.389i)2-s + (−0.809 − 0.587i)3-s + (0.696 + 0.717i)4-s + (−0.254 − 0.967i)5-s + (0.516 + 0.856i)6-s + (0.941 − 0.336i)7-s + (−0.362 − 0.931i)8-s + (0.309 + 0.951i)9-s + (−0.142 + 0.989i)10-s + (−0.142 − 0.989i)12-s + (0.897 − 0.441i)13-s + (−0.998 − 0.0570i)14-s + (−0.362 + 0.931i)15-s + (−0.0285 + 0.999i)16-s + (−0.736 − 0.676i)17-s + (0.0855 − 0.996i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.708 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.708 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1932351331 - 0.4673686919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1932351331 - 0.4673686919i\) |
\(L(1)\) |
\(\approx\) |
\(0.4612629887 - 0.3369334112i\) |
\(L(1)\) |
\(\approx\) |
\(0.4612629887 - 0.3369334112i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.921 - 0.389i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.254 - 0.967i)T \) |
| 7 | \( 1 + (0.941 - 0.336i)T \) |
| 13 | \( 1 + (0.897 - 0.441i)T \) |
| 17 | \( 1 + (-0.736 - 0.676i)T \) |
| 19 | \( 1 + (0.198 - 0.980i)T \) |
| 23 | \( 1 + (-0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.870 - 0.491i)T \) |
| 31 | \( 1 + (-0.985 + 0.170i)T \) |
| 37 | \( 1 + (-0.466 - 0.884i)T \) |
| 41 | \( 1 + (0.974 - 0.226i)T \) |
| 43 | \( 1 + (0.841 + 0.540i)T \) |
| 47 | \( 1 + (0.0855 + 0.996i)T \) |
| 53 | \( 1 + (-0.0285 - 0.999i)T \) |
| 59 | \( 1 + (0.974 + 0.226i)T \) |
| 61 | \( 1 + (-0.921 + 0.389i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
| 71 | \( 1 + (0.993 - 0.113i)T \) |
| 73 | \( 1 + (0.610 - 0.791i)T \) |
| 79 | \( 1 + (0.774 + 0.633i)T \) |
| 83 | \( 1 + (-0.564 + 0.825i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.254 + 0.967i)T \) |
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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
−29.14302628267154423089067746676, −28.07381762762063347285980043067, −27.4725669535088552692039012391, −26.53520503646078925381343583231, −25.79236432529473286166729469854, −24.266955519830098660254255296213, −23.542307892697010439730630179466, −22.374899027915883795777094175879, −21.27969226202867812908175085462, −20.214374719602595670701284506445, −18.590874013900012960812915127684, −18.20039488006493834067934061685, −17.11907023636636473607400824262, −16.00645610687338286146806853466, −15.13509758243648751419600033104, −14.28062580425562121446047294253, −11.96595304765338365359575488816, −11.0515499100789941312220844517, −10.47551795841018673369747277111, −9.07870027650685218903615884876, −7.83800788353144073347389292666, −6.51186124185613961975917063611, −5.62205677432233550116172496089, −3.9230532576510951559611670450, −1.82634204411086144773365863019,
0.73395664596134848507534692301, 1.95508961793995449784921131693, 4.216883165395565812270994723823, 5.64558756787515785331625966266, 7.25133227904168566163209290854, 8.09875681700026930918405377987, 9.24834114225403571272325200841, 10.9216197874605945744546979140, 11.432611032744238490413817378106, 12.58838080460774859115975386479, 13.55633921363147313741969214137, 15.70346009522532184013821247138, 16.46870017200901985238628583223, 17.70391735799398353757958270042, 17.95761477519117244442638198606, 19.44171943238645502699228646504, 20.34533519311228506661899835110, 21.242650596506895520384925991503, 22.59092789814877841905465605817, 24.05282773178791451790800164562, 24.40564298074240844082321395681, 25.65585530151058381099560797382, 27.07208995652040763135518523645, 27.92142291942389850758621932297, 28.40050290498239687721619949888