L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s − 15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s − 15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7831311043 - 0.5209246280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7831311043 - 0.5209246280i\) |
\(L(1)\) |
\(\approx\) |
\(0.9853529735 - 0.3949586307i\) |
\(L(1)\) |
\(\approx\) |
\(0.9853529735 - 0.3949586307i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−33.16506596328657197526284612113, −32.14484342853559572918264277117, −31.019623498325764102500489575255, −30.1945458082384610839988209397, −28.53313082223360704264976242599, −27.43366908793210430303059774234, −26.37753373264914409632099503886, −25.80149430434257479640600552011, −24.08505356587280379312737784967, −22.78749585396203842608140817548, −21.7569889886932054639833834159, −20.6940901781594864447242360442, −19.38365351217698016445250029531, −18.40955407094365035674089624407, −16.5811691152419186555591431359, −15.543349844025600373333509479349, −14.57460036805624808450531222602, −13.33677315490547045157807133177, −11.24474540088059881194093773060, −10.539992279648797561605690132891, −8.91619288649665427146502927957, −7.698173118292260187384135206786, −5.84367434213653341480433658441, −3.98274788399003067204061816101, −2.86592229947028444769445107760,
1.48445924299526608427941871755, 3.49915442338122875007197382990, 5.37105154696096913402046805253, 7.222322751380498019729854429015, 8.24030201410243590146529431667, 9.515195944154144649061963124194, 11.589505532440730239131325453577, 12.65598227578844444948413991532, 13.64193665242741608862021068466, 15.140616087304156406842556961449, 16.41450138176173723125088805570, 17.91090697953854904828455114495, 18.9201023277689602059161126016, 20.269692987560820180159614551004, 20.81798067798255846963686266251, 23.00688953255070787106684640005, 23.69484806656172175541247167264, 24.954033714979342472829084135097, 25.71538626442790863862629935448, 27.25379731544374331712185001285, 28.44834213985124964583053717317, 29.45440801738142322946192450052, 30.88534796706838815003140710171, 31.40354834054244039757311782718, 32.62226581537803238142745106992