L(s) = 1 | + (0.793 + 0.608i)2-s + (0.991 − 0.130i)3-s + (0.258 + 0.965i)4-s + (−0.751 + 0.659i)5-s + (0.866 + 0.5i)6-s + (0.946 − 0.321i)7-s + (−0.382 + 0.923i)8-s + (0.965 − 0.258i)9-s + (−0.997 + 0.0654i)10-s + (0.608 + 0.793i)11-s + (0.382 + 0.923i)12-s + (−0.659 − 0.751i)13-s + (0.946 + 0.321i)14-s + (−0.659 + 0.751i)15-s + (−0.866 + 0.5i)16-s + (−0.321 + 0.946i)17-s + ⋯ |
L(s) = 1 | + (0.793 + 0.608i)2-s + (0.991 − 0.130i)3-s + (0.258 + 0.965i)4-s + (−0.751 + 0.659i)5-s + (0.866 + 0.5i)6-s + (0.946 − 0.321i)7-s + (−0.382 + 0.923i)8-s + (0.965 − 0.258i)9-s + (−0.997 + 0.0654i)10-s + (0.608 + 0.793i)11-s + (0.382 + 0.923i)12-s + (−0.659 − 0.751i)13-s + (0.946 + 0.321i)14-s + (−0.659 + 0.751i)15-s + (−0.866 + 0.5i)16-s + (−0.321 + 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0711 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0711 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.566377588 + 2.389804748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.566377588 + 2.389804748i\) |
\(L(1)\) |
\(\approx\) |
\(1.910776600 + 1.032425283i\) |
\(L(1)\) |
\(\approx\) |
\(1.910776600 + 1.032425283i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (0.793 + 0.608i)T \) |
| 3 | \( 1 + (0.991 - 0.130i)T \) |
| 5 | \( 1 + (-0.751 + 0.659i)T \) |
| 7 | \( 1 + (0.946 - 0.321i)T \) |
| 11 | \( 1 + (0.608 + 0.793i)T \) |
| 13 | \( 1 + (-0.659 - 0.751i)T \) |
| 17 | \( 1 + (-0.321 + 0.946i)T \) |
| 19 | \( 1 + (0.195 + 0.980i)T \) |
| 23 | \( 1 + (-0.442 - 0.896i)T \) |
| 29 | \( 1 + (0.997 + 0.0654i)T \) |
| 31 | \( 1 + (-0.130 - 0.991i)T \) |
| 37 | \( 1 + (0.896 + 0.442i)T \) |
| 41 | \( 1 + (0.0654 - 0.997i)T \) |
| 43 | \( 1 + (-0.965 - 0.258i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.608 - 0.793i)T \) |
| 59 | \( 1 + (0.442 - 0.896i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.980 - 0.195i)T \) |
| 71 | \( 1 + (0.0654 + 0.997i)T \) |
| 73 | \( 1 + (0.258 - 0.965i)T \) |
| 79 | \( 1 + (-0.923 - 0.382i)T \) |
| 83 | \( 1 + (-0.946 - 0.321i)T \) |
| 89 | \( 1 + (0.382 - 0.923i)T \) |
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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
−29.94478091343079207256961130534, −28.61860864015568347989951951716, −27.444539004459006899652749251, −26.86315484247231050995654570822, −24.89516885607498277613864617290, −24.40304189187732333762072709001, −23.49279988221505508200225647354, −21.75431056525706574590315659411, −21.297645472117740126977647979508, −19.89784532280111705658308486347, −19.64820653131238338799007536066, −18.273930226758349578883473281362, −16.28071132827626755176530509934, −15.28555674096919913723741550776, −14.28713298404360560757866576983, −13.44451496819156849577055704101, −11.99072212744217144008061397045, −11.30379255980877432432419644139, −9.50648099402571333890149571917, −8.568068032772781256454196164946, −7.107664783971295561400270444557, −5.028313485623048409641205773709, −4.190418725850727422444768882016, −2.792872056934194360186715710630, −1.28241226385783953275788096861,
2.20201023800986516795992536586, 3.6813945527209434745507590745, 4.57070528304514631358299543447, 6.58930388684537665845805908099, 7.711393575848332159575496806768, 8.287907921680189117081697609088, 10.262296889249747401376221738, 11.8209561542628999583622152159, 12.81791572621635047876260892647, 14.39302006513342960675610756268, 14.66703479643336263704629721463, 15.61249914245203671147752384301, 17.18386153294650387009592571964, 18.33097273972279175613960688146, 19.85157376892391103099553703394, 20.51989105788541249300822887058, 21.82695555525794537421086864821, 22.858452404326433515161042399274, 23.97008855248296281879004455796, 24.7624122117959775042844061484, 25.80076723638343334010511896091, 26.801457250140287219844255069368, 27.48500615904997898110398250727, 29.70872753674888973451405654442, 30.65488685718328289924048366148