| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.186731825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.186731825\) |
| \(L(1)\) |
\(\approx\) |
\(3.116259089\) |
| \(L(1)\) |
\(\approx\) |
\(3.116259089\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1129 | \( 1 \) |
| 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
−21.44249665855035467453808242120, −20.65615348710683553939169209176, −20.2301371720108592429323651309, −19.17274449517335592627433727982, −18.35159744284517854654249529520, −17.374603886444961683973907484738, −16.667905688679408128445731048648, −15.31970302542411591153203485667, −15.0563834562685658283242111180, −14.26284012286780859791508413389, −13.55482826539730468883162736743, −12.98957741191263150138293924454, −12.2645028265479487250620391841, −10.81576930885260128407860615074, −10.56804839195669176557610597202, −9.3309186383717471271416340025, −8.51395304621024251643629872283, −7.45763636138802843480077874287, −6.93074726724482807742891366004, −5.62491971447661710095710980492, −4.93105973836882693701557622212, −4.20536700470735074299804884191, −2.90187934514116238267179044881, −2.21868698960341199890276093058, −1.67983703749849900230490876660,
1.67983703749849900230490876660, 2.21868698960341199890276093058, 2.90187934514116238267179044881, 4.20536700470735074299804884191, 4.93105973836882693701557622212, 5.62491971447661710095710980492, 6.93074726724482807742891366004, 7.45763636138802843480077874287, 8.51395304621024251643629872283, 9.3309186383717471271416340025, 10.56804839195669176557610597202, 10.81576930885260128407860615074, 12.2645028265479487250620391841, 12.98957741191263150138293924454, 13.55482826539730468883162736743, 14.26284012286780859791508413389, 15.0563834562685658283242111180, 15.31970302542411591153203485667, 16.667905688679408128445731048648, 17.374603886444961683973907484738, 18.35159744284517854654249529520, 19.17274449517335592627433727982, 20.2301371720108592429323651309, 20.65615348710683553939169209176, 21.44249665855035467453808242120
