L(s) = 1 | + (−0.623 − 0.781i)2-s + (0.330 − 0.943i)3-s + (−0.222 + 0.974i)4-s + (−0.330 − 0.943i)5-s + (−0.943 + 0.330i)6-s + (−0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (−0.532 + 0.846i)10-s + (−0.974 + 0.222i)11-s + (0.846 + 0.532i)12-s + (−0.433 − 0.900i)13-s + (0.900 + 0.433i)14-s − 15-s + (−0.900 − 0.433i)16-s + (0.993 + 0.111i)17-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (0.330 − 0.943i)3-s + (−0.222 + 0.974i)4-s + (−0.330 − 0.943i)5-s + (−0.943 + 0.330i)6-s + (−0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (−0.532 + 0.846i)10-s + (−0.974 + 0.222i)11-s + (0.846 + 0.532i)12-s + (−0.433 − 0.900i)13-s + (0.900 + 0.433i)14-s − 15-s + (−0.900 − 0.433i)16-s + (0.993 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.08028866876 - 0.4576198386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.08028866876 - 0.4576198386i\) |
\(L(1)\) |
\(\approx\) |
\(0.3804526868 - 0.4693711288i\) |
\(L(1)\) |
\(\approx\) |
\(0.3804526868 - 0.4693711288i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 113 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 3 | \( 1 + (0.330 - 0.943i)T \) |
| 5 | \( 1 + (-0.330 - 0.943i)T \) |
| 7 | \( 1 + (-0.900 + 0.433i)T \) |
| 11 | \( 1 + (-0.974 + 0.222i)T \) |
| 13 | \( 1 + (-0.433 - 0.900i)T \) |
| 17 | \( 1 + (0.993 + 0.111i)T \) |
| 19 | \( 1 + (-0.943 - 0.330i)T \) |
| 23 | \( 1 + (0.943 - 0.330i)T \) |
| 29 | \( 1 + (-0.993 - 0.111i)T \) |
| 31 | \( 1 + (0.433 + 0.900i)T \) |
| 37 | \( 1 + (0.532 - 0.846i)T \) |
| 41 | \( 1 + (-0.974 - 0.222i)T \) |
| 43 | \( 1 + (0.111 - 0.993i)T \) |
| 47 | \( 1 + (0.846 - 0.532i)T \) |
| 53 | \( 1 + (0.222 - 0.974i)T \) |
| 59 | \( 1 + (0.943 + 0.330i)T \) |
| 61 | \( 1 + (0.974 - 0.222i)T \) |
| 67 | \( 1 + (-0.846 - 0.532i)T \) |
| 71 | \( 1 + (0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 + (-0.532 - 0.846i)T \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.111 - 0.993i)T \) |
| 97 | \( 1 + (-0.900 - 0.433i)T \) |
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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
−29.689699165890558454925392278219, −28.68242712011277866668025083862, −27.49743277459007830038260345072, −26.66714900667587212397385102546, −26.05246881228614713616017855433, −25.395748003959103934455993818806, −23.640186710662394640036758790053, −22.96763141470526956015855238411, −21.87239852115335617973548508256, −20.545115916781598738593912620959, −19.13862674826660623750125761982, −18.85458252228046909462839700597, −17.0443549285958486630861808916, −16.32930300611126660588859366235, −15.29617565963200803635910596156, −14.55279185354559743535036678262, −13.42142778175823003479785884498, −11.21901600524772936384654938648, −10.23998537260842198011506048657, −9.544010976341065159071731230343, −8.09580449431634128814451844978, −7.03608671510722730328175363088, −5.73662050611423354350398440177, −4.18853819713499077118240535878, −2.75499895429718495642106622042,
0.50641649341030994244022848258, 2.25238369765044463702786916785, 3.412016537131324187598971561150, 5.35652427657658312905815843115, 7.2048829228124353636924167368, 8.22534299923429012024920666837, 9.09702206469322149952504345931, 10.37982048970252845088690306189, 12.00242238418008882636925843596, 12.75985157816939352795590096415, 13.21070349518407330842711983993, 15.20743633897383715828806751208, 16.54789502340291474410593887625, 17.52701041464412620720479633183, 18.72784160439452943967414816437, 19.38404423531395210768406366218, 20.31110085910757421169473862956, 21.17843126947092920330584660713, 22.717260539352723101741141569594, 23.667435196907458976543935520552, 25.133011076431885759148147150323, 25.5633879449187219389654906868, 26.863601497565526143811820692530, 28.16164435110967719188007609581, 28.75633647808961094086169514042