| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 929 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 929 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.455267971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455267971\) |
| \(L(1)\) |
\(\approx\) |
\(1.693910206\) |
| \(L(1)\) |
\(\approx\) |
\(1.693910206\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 929 | \( 1 \) |
| 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
−21.90503528688039722606826733819, −21.68049327903127072396448167116, −20.34085588076505474824878093597, −19.72476909931962008751637760330, −18.69822866076576482082245504782, −17.65311224586414197060963150420, −16.88192723726774896134653829752, −16.46119358193829757850415933169, −15.463023734332907104714521548001, −14.641992596238627484894670832731, −13.608383936334281488063604774365, −13.07570139062545382335946902664, −12.262811775223803160626040962, −11.60949265958652558341747787894, −10.60647309546776013540627413452, −9.87299245937357474732681780906, −9.10956900609631016351674019628, −7.16944856586520800221899618023, −6.764590470363471852583696398042, −6.014968040865106998868968415366, −5.21614548225621439054523246813, −4.44327128724878020669524139503, −3.28042616790224897711922160463, −2.245546332508404026545812323720, −1.10268568255822586687285538372,
1.10268568255822586687285538372, 2.245546332508404026545812323720, 3.28042616790224897711922160463, 4.44327128724878020669524139503, 5.21614548225621439054523246813, 6.014968040865106998868968415366, 6.764590470363471852583696398042, 7.16944856586520800221899618023, 9.10956900609631016351674019628, 9.87299245937357474732681780906, 10.60647309546776013540627413452, 11.60949265958652558341747787894, 12.262811775223803160626040962, 13.07570139062545382335946902664, 13.608383936334281488063604774365, 14.641992596238627484894670832731, 15.463023734332907104714521548001, 16.46119358193829757850415933169, 16.88192723726774896134653829752, 17.65311224586414197060963150420, 18.69822866076576482082245504782, 19.72476909931962008751637760330, 20.34085588076505474824878093597, 21.68049327903127072396448167116, 21.90503528688039722606826733819
