| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.765808459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.765808459\) |
| \(L(1)\) |
\(\approx\) |
\(1.227001578\) |
| \(L(1)\) |
\(\approx\) |
\(1.227001578\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 \) |
| 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 | \( 1 + T \) |
| 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
−32.62597506746700614006662791729, −31.19958303885995070219023580384, −30.0324010196726329332126754077, −29.16100365280325938271044606275, −27.79011163666017973039843257895, −26.68094775485721075152939190393, −25.87806186021124020711720564769, −24.82402946979965173748262956374, −24.08601145647857941057807043640, −21.569070605424126618547423808772, −20.896101243085990628597678308829, −19.88682078807673111459341478801, −18.44399875207251902365075412344, −17.790219758596701721785145725428, −16.32249866535919391820100029157, −14.89651083104486534629573970204, −13.93416825418026727211224589156, −12.26165300698673870099690387165, −10.41842848762886878007922085459, −9.61712931549233611956560246964, −8.25421262721750835118876189276, −7.32539165541390632203700205008, −5.32387734981457323422720248811, −2.77335914537658608384341778358, −1.60645524674271843040065330326,
1.60645524674271843040065330326, 2.77335914537658608384341778358, 5.32387734981457323422720248811, 7.32539165541390632203700205008, 8.25421262721750835118876189276, 9.61712931549233611956560246964, 10.41842848762886878007922085459, 12.26165300698673870099690387165, 13.93416825418026727211224589156, 14.89651083104486534629573970204, 16.32249866535919391820100029157, 17.790219758596701721785145725428, 18.44399875207251902365075412344, 19.88682078807673111459341478801, 20.896101243085990628597678308829, 21.569070605424126618547423808772, 24.08601145647857941057807043640, 24.82402946979965173748262956374, 25.87806186021124020711720564769, 26.68094775485721075152939190393, 27.79011163666017973039843257895, 29.16100365280325938271044606275, 30.0324010196726329332126754077, 31.19958303885995070219023580384, 32.62597506746700614006662791729
