| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s + 20-s − 21-s − 22-s − 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s + 20-s − 21-s − 22-s − 23-s − 24-s + 25-s + 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 661 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 661 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.508256920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508256920\) |
| \(L(1)\) |
\(\approx\) |
\(1.119993251\) |
| \(L(1)\) |
\(\approx\) |
\(1.119993251\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 661 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 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
−22.62388968944752790378195159599, −21.64163106719771347674648156909, −21.10435796179959937762040297158, −20.10555918497293237513767721945, −19.355629684403533687241675289104, −19.04282347174214970068403323662, −17.87846131849304140909684246993, −17.10033376875903115573378346989, −16.38876684561043376716702107862, −15.430537774437011560970788367224, −14.47872533418285009482351757966, −13.889650888797668313902386438035, −12.59969459025008894706061339224, −12.1107197462051398732158428718, −10.3136011444571491290598630759, −10.03760136877230371192855176121, −9.2266273525897068636295543419, −8.595271630125436502569595626840, −7.42281956466655232075327137243, −6.637686693420421010399258940287, −5.85129764673081206993223295351, −4.13794117650138357053891931904, −2.9120320855430778369792135490, −2.26993119468808849567455376034, −1.13821843128697013398837190645,
1.13821843128697013398837190645, 2.26993119468808849567455376034, 2.9120320855430778369792135490, 4.13794117650138357053891931904, 5.85129764673081206993223295351, 6.637686693420421010399258940287, 7.42281956466655232075327137243, 8.595271630125436502569595626840, 9.2266273525897068636295543419, 10.03760136877230371192855176121, 10.3136011444571491290598630759, 12.1107197462051398732158428718, 12.59969459025008894706061339224, 13.889650888797668313902386438035, 14.47872533418285009482351757966, 15.430537774437011560970788367224, 16.38876684561043376716702107862, 17.10033376875903115573378346989, 17.87846131849304140909684246993, 19.04282347174214970068403323662, 19.355629684403533687241675289104, 20.10555918497293237513767721945, 21.10435796179959937762040297158, 21.64163106719771347674648156909, 22.62388968944752790378195159599
