L(s) = 1 | + (−0.5 − 0.866i)3-s − i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s − i·31-s + (0.866 + 0.5i)33-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s − i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s − i·31-s + (0.866 + 0.5i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1560323651 - 0.5536621276i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1560323651 - 0.5536621276i\) |
\(L(1)\) |
\(\approx\) |
\(0.5814863469 - 0.4018925901i\) |
\(L(1)\) |
\(\approx\) |
\(0.5814863469 - 0.4018925901i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−29.97574274234549050496287520285, −29.071839478877398782122184432576, −28.165416634253221227371492352652, −27.02364992175032501083092440332, −26.15770567866800981009505875808, −25.432196860376878880245887226665, −23.54732108351929339185727939668, −22.91440917153911113515848401987, −21.73196424752087227953853023806, −21.3111867130420794232885041588, −19.578608722957906113931708262475, −18.63828951390590166070728047368, −17.499002826513931198000213957484, −16.20664564450725413984525730468, −15.43436247094649590093397388822, −14.427771801131092353681869586906, −12.890217934224119813018656800842, −11.57761725964315416240920412144, −10.469187541449411133326447355345, −9.76327653064516038335531572822, −8.18590136067416860058057106353, −6.41837475287374512147613335455, −5.65673199873774513842900221314, −3.85474843868895714007407009780, −2.770043114085227843898850474436,
0.59036288290125460867570716603, 2.43133252982862386566356446494, 4.49784270075083894544161553131, 5.74758110102134486210934216699, 7.03892612965073740267861882848, 8.13490685170703307236557897004, 9.5905041282896414643229906693, 10.90869582437913371392014667537, 12.44753181784409319611108750494, 12.863510261365110123313360640254, 13.981239824827666908956567331628, 15.87306335334387743441610289649, 16.65453153536574798548380049213, 17.66554537191525321832197584129, 18.81615663907485363260478295487, 19.8748983133599150790360863832, 20.785871042436299784493349870705, 22.33132610907509956757701436637, 23.34773342724818233058810126783, 23.9840409349194026338015772219, 25.17463121134519193857197204388, 25.97728453780237386825233716438, 27.582047622218384172331683855306, 28.56091165894024284607059949985, 29.19003146195579568140497110229